Prednášky MA1

Prednášky z Matematickej analýzy 1

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Základné pojmy

Kvantifikátory, sumátory, dôkazy v matematike (priamy, nepriamy, sporom, matematickou indukciou), množiny a operácie s množinami (prienik, zjednotenie, rozdiel, doplnok), zobrazenia množín (injekcia, surjekcia, bijekcia), zložené zobrazenie, mohutnosť množín, operácie s nekonečnom, intervaly, okolie bodu, otvorené a uzavreté množiny.

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Číselné postupnosti

Vlastnosti (ohraničenosť, monotónnosť), podpostupnosti, hromadná hodnota, limes superior, limes inferior, limita, konvergencia, divergencia, dôležité limity.

Riešené limity:  $01.\,\lim\limits_{n\to\infty}{\frac{n^2+n}{n^2-2}}.\enspace$ $02.\,\lim\limits_{n\to\infty}{n\big[\sqrt[n]3-\sqrt[n]2\big]}.\enspace$ $03.\,\lim\limits_{n\to\infty}{\frac{n^3-2}{n^2+n}}.\enspace$ $04.\,\lim\limits_{n\to\infty}{\frac{n^2+n}{n^4-n^3}}.\enspace$ $05.\,\lim\limits_{n\to\infty}{\big[\sqrt{n+1}-n\big]}.\enspace$ $06.\,\lim\limits_{n\to\infty}{\big[\sqrt{n+1}-\sqrt{n}\big]}.\enspace$ $07.\,\lim\limits_{n\to\infty}{\frac{2^n+3^n}{2^{n+1}+3^{n+1}}}.\enspace$ $08.\,\lim\limits_{n\to\infty}{\frac{(-2)^n+(-3)^n}{(-2)^{n+1}+(-3)^{n+1}}}.\enspace$ $09.\,\lim\limits_{n\to\infty}{\big(1+\frac1n\big)^{2n}}.\enspace$ $10.\,\lim\limits_{n\to\infty}{\big(1+\frac1n\big)^{n^2+1}}.\enspace$ $11.\,\lim\limits_{n\to\infty}{\big(1+\frac1n\big)^{\sqrt{n}+1}}.\enspace$ $12.\,\lim\limits_{n\to\infty}{\big[\frac{n}{2}-\frac{1+2+3+\cdots+n}{n+2}\big]}.\enspace$ $13.\,\lim\limits_{n\to\infty}{\sqrt[n]{3^n-2^n}}.\enspace$ $14.\,\lim\limits_{n\to\infty}{\sqrt[n]{2^n+1}}.\enspace$ $15.\,\lim\limits_{n\to\infty}{\frac{1-\sqrt{n}}{1+\sqrt{n}}}.\enspace$ $16.\,\lim\limits_{n\to\infty}{n\big[\sqrt[n]{2}-\sqrt[n]{3}\big]}.\enspace$ $17.\,\lim\limits_{n\to\infty}{\big[\frac{n^2}{n+2}-\frac{n^2}{n+3}\big]}.\enspace$ $18.\,\lim\limits_{n\to\infty}{\frac{n+3^n}{n-3^n}}.\enspace$ $19.\,\lim\limits_{n\to\infty}{\frac{n+3^{-n}}{n-3^{-n}}}.\enspace$ $20.\,\lim\limits_{n\to\infty}{\big[\sqrt[3]{n^2+3n+1}-\sqrt[3]{n^2+n+2}\big]}.\enspace$ $21.\,\lim\limits_{n\to\infty}{4^{\frac{8n+1}{4n-3}}}.\enspace$ $22.\,\lim\limits_{n\to\infty}{\big(\frac{2n-1}{2n+3}\big)^6}.\enspace$ $23.\,\lim\limits_{n\to\infty}{\frac{1+2+3+\cdots+n}{\sqrt{9n^4+1}}}.\enspace$ $24.\,\lim\limits_{n\to\infty}{\frac{1+3+5+\cdots+(2n-1)}{1+2+3+\cdots+n}}.\enspace$ $25.\,\lim\limits_{n\to\infty}{\frac{1^2+2^2+3^2+\cdots+n^2}{n^3}}.\enspace$ $26.\,\lim\limits_{n\to\infty}{\frac{1}{\sqrt{n^2+n+1}-\sqrt{n^2+n+2}}}.\enspace$ $27.\,\lim\limits_{n\to\infty}{\frac{1^2+3^2+5^2+\cdots+(2n-1)^2}{n^3}}.\enspace$ $28.\,\lim\limits_{n\to\infty}{\frac{1}{\sqrt{n^2-n+1}-\sqrt{n^2-n-1}}}.\enspace$ $29.\,\lim\limits_{n\to\infty}{n\big[\ln{n}-\ln{(n+2)}\big]}.\enspace$ $30.\,\lim\limits_{n\to\infty}{n\big[\ln{(n+3)}-\ln{n}\big]}.\enspace$ $31.\,\lim\limits_{n\to\infty}{\big[\sqrt{n^2+n+1}-\sqrt{n^2+n+2}\big]}.\enspace$ $32.\,\lim\limits_{n\to\infty}{\frac{8+3^{-n}+4\cdot5^{-n}}{3n+2-n\cdot2^{-n}}}.\enspace$ $33.\,\lim\limits_{n\to\infty}{\big[\sqrt{n^2-n+1}-\sqrt{n^2-3n+2}\big]}.\enspace$ $34.\,\lim\limits_{n\to\infty}{\big[\sqrt[3]{n+1}-\sqrt[3]{n+2}\big]}.\enspace$ $35.\,\lim\limits_{n\to\infty}{\frac{1}{\sqrt[3]{n-1}-\sqrt[3]{n-2}}}.\enspace$ $36.\,\lim\limits_{n\to\infty}{\big[\sqrt[4]{n^4-1}-\sqrt[4]{n^4+1}\big]}.\enspace$ $37.\,\lim\limits_{n\to\infty}{\big[\sqrt{n^2+4n+1}-n+1\big]}.\enspace$ $38.\,\lim\limits_{n\to\infty}{\big[\sqrt{n^2+n+1}-\sqrt{n-1}\big]}.\enspace$ $39.\,\lim\limits_{n\to\infty}{\frac{1}{\sqrt[3]{n^3+n+1}-\sqrt[3]{n-1}}}.\enspace$ $40.\,\lim\limits_{n\to\infty}{\big[\sqrt[4]{n^4+1}-\sqrt[4]{n+1}\big]}.\enspace$ $41.\,\lim\limits_{n\to\infty}{\frac{1}{\sqrt[3]{n^3+1}-n}}.\enspace$ $42.\,\lim\limits_{n\to\infty}{\frac{1}{\sqrt[5]{n^5+1}-\sqrt[5]{n^4+1}}}.\enspace$ $43.\,\lim\limits_{n\to\infty}{\big[\sqrt[3]{n^3+1}-n+1\big]}.\enspace$ $44.\,\lim\limits_{n\to\infty}{\big[\sqrt[3]{n^3+1}-n+1\big]}.\enspace$ $45.\,\lim\limits_{n\to\infty}{\big(\frac{2n-1}{2n+3}\big)^{n+6}}.\enspace$ $46.\,\lim\limits_{n\to\infty}{\frac{2n^3+3n^4-n^2+2\cdot5^n}{2n^2-n^3+3\cdot5^{n+1}}}.\enspace$ $47.\,\lim\limits_{n\to\infty}{\big(\frac{2n-1}{2n+3}\big)^{n^6+6}}.\enspace$ $48.\,\lim\limits_{n\to\infty}{\frac{2n^3+3n^4-n^2}{2n^2-n^3+3\cdot5^{n+1}}}.\enspace$ $49.\,\lim\limits_{n\to\infty}{\big(\frac{2n-1}{2n-3}\big)^{n^6+6}}.\enspace$ $50.\,\lim\limits_{n\to\infty}{\sqrt[n]{\frac{5n+1}{n+5}}}.\enspace$ $51.\,\lim\limits_{n\to\infty}{\big(\frac{2n^6-1}{2n^6+3}\big)^{n+6}}.\enspace$ $52.\,\lim\limits_{n\to\infty}{\frac{2n^2+3n^3+5}{n^3-n^4-n^2+2}}.\enspace$ $53.\,\lim\limits_{n\to\infty}{\big(\frac{2n^6-1}{2n^6+3}\big)^{n^6+6}}.\enspace$ $54.\,\lim\limits_{n\to\infty}{\frac{n^3-n^4+2n^2+2}{2n^2-3n^3+5}}.\enspace$ $55.\,\lim\limits_{n\to\infty}{\big(\frac{2\sqrt[6]{n}-1}{2\sqrt[6]{n}+3}\big)^{\sqrt[6]{n}+6}}.\enspace$ $56.\,\lim\limits_{n\to\infty}{\frac{2n^2+3n^4+5}{n^3-n^4-n^2+2}}.\enspace$ $57.\,\lim\limits_{n\to\infty}{\big(\frac{2n-1}{2n+3}\big)^{\sqrt[6]{n}+6}}.\enspace$ $58.\,\lim\limits_{n\to\infty}{\frac{2^n.\,n!}{n^n}}.\enspace$ $59.\,\lim\limits_{n\to\infty}{\big(\frac{2\sqrt[6]{n}-1}{2\sqrt[6]{n}+3}\big)^{n+6}}.\enspace$ $60.\,\lim\limits_{n\to\infty}{\frac{3^n.\,n!}{n^n}}.\enspace$ $61.\,\lim\limits_{n\to\infty}{\big(\frac{2\sqrt[6]{n}-1}{2\sqrt[6]{n}-3}\big)^{n+6}}.\enspace$ $62.\,\lim\limits_{n\to\infty}{\frac{8^n.\,n!}{(3n)^n}}.\enspace$ $63.\,\lim\limits_{n\to\infty}{\frac{2n^3+3n^4-n^2-2\cdot5^n}{2n^2-n^3}}.\enspace$ $64.\,\lim\limits_{n\to\infty}{\frac{2n^3+3n^4-n^2+2\cdot5^n}{2n^2-n^3+6\cdot7^n}}.\enspace$ $65.\,\lim\limits_{n\to\infty}{\frac{\sqrt[3]{n^4-1}-\sqrt[5]{n^6-2}+3\sqrt{n+1}}{2\sqrt[3]{n^5+1}+3\sqrt[4]{n^6-1}-\sqrt{n-1}}}.\enspace$ $66.\,\lim\limits_{n\to\infty}{\frac{(\frac13)^n}{(\frac15)^n-(\frac14)^n}}.\enspace$ $67.\,\lim\limits_{n\to\infty}{\frac{(\frac14)^n}{(\frac15)^n-(\frac13)^n}}.\enspace$ $68.\,\lim\limits_{n\to\infty}{\frac{(\frac15)^n}{(\frac14)^n-(\frac13)^n}}.\enspace$ $69.\,\lim\limits_{n\to\infty}{\frac{2n^3+3n^4-n^2+2\cdot5^n}{2n^2-n^3}}.\enspace$ $70.\,\lim\limits_{n\to\infty}{\frac{\sqrt{n}}{\sqrt{n+\sqrt{n+\sqrt{n}}}}}.\enspace$ $71.\,\lim\limits_{n\to\infty}{\frac{2\sqrt[3]{n}+3\sqrt[4]{n}-\sqrt{n}}{2\sqrt{n}-\sqrt[5]{n}+3}}.\enspace$ $72.\,\lim\limits_{n\to\infty}{\big(\frac{\sqrt[n]{2}+\sqrt[n]{3}}{2}\big)^n}.\enspace$ $73.\,\lim\limits_{n\to\infty}{n\left[\sqrt[n]{2}-\sqrt[n+1]{2}.\,\right]}.\enspace$ $74.\,\lim\limits_{n\to\infty}{n^2\big[\sqrt[n]{2}-\sqrt[n+1]{2}.\,\big]}.\enspace$ $75.\,\lim\limits_{n\to\infty}{\frac1n\ln{\big[2^n+\sqrt{2^n}+\sqrt[3]{2^n}+\cdots+\sqrt[n]{2^n}\big]}}.\enspace$ $76.\,\lim\limits_{n\to\infty}{\frac{2n^3+3n^4-n^2+2\cdot5^n}{2n^2-n^3+6\cdot4^n}}.\enspace$ $77.\,\lim\limits_{n\to\infty}{a_n}\text{ pre }\{a_n\}_{n=1}^{\infty}=\Big\{\sqrt2,\sqrt{\sqrt2},\sqrt{\sqrt{\sqrt2}},\dots\Big\}.\enspace$ $78.\,\lim\limits_{n\to\infty}{a_n}\text{ pre }\{a_n\}_{n=1}^{\infty}=\Big\{\sqrt2,\sqrt{2+\sqrt2},\sqrt{2+\sqrt{2+\sqrt2}},\dots\Big\}.\enspace$ $79.\,\lim\limits_{n\to\infty}{a_n},\text{ ak }a_1=2,\ a_{n+1}=\sqrt{2a_n+3},\ n\in N.\enspace$ $80.\,\text{Vyjadrite }32,17\overline{71}\text{ ako zlomok}.$

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Číselné rady

Postupnosť čiastočných súčtov, konvergencia, divergencia, harmonický rad, nutná podmienka konvergencie, kritéria konvergencie radov (porovnávacie, D'Alembertovo, Cauchyho) a ich limitné tvary, dôležité rady, riešené príklady, alternujúce rady, absolútna a relatívna konvergencia, Leibnizovo kritérium.

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Reálne funkcie reálnej premennej

Explicitný, parametrický a implicitný tvar, operácie s funkciami, Dirichletova funkcia, lokálne a globálne vlastnosti, ohraničenosť, extrémy, infimum a suprémum funkcie, monotónnosť, párnosť a nepárnosť, periodickosť, konvexnosť a konkávnosť, skladanie funkcií, inverzná funkcia.

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Elementárne funkcie

Elementárne funkcie a ich vlastnosti, polynóm, racionálna lomená funkcia, mocninná funkcia, exponenciálna funkcia, logaritmická funkcia, goniometrické funkcie (sínus, kosínus, tangens, kotangens), cyklometrické funkcie (arkussínus, arkuskosínus, arkustangens, arkuskotangens), hyperbolické funkcie (sínus hyperbolický, kosínus hyperbolický, tangens hyperbolický, kotangens hyperbolický), hyperbolometrické funkcie (argument sínus hyperbolický, argument kosínus hyperbolický, argument tangens hyperbolický, argument kotangens hyperbolický).

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Limita funkcie

Definícia a základné vlastnosti, dôležité limity, jednostranné limity, riešené príklady.

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Spojitosť funkcie

Spojitosť funkcie v bode a na množine, definícia, spojitosť s limitou funkcie, nespojitosť (odstrániteľná a neodstrániteľná I., II. druhu), spojitosť na uzavretom intervale, metóda bisekcie.

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Derivácia funkcie

Geometrická interpretácia, obojstranná a jednostranné derivácie, derivácie elementárnych funkcií, vzťah medzi spojitosťou a deriváciou, základné vzorce pre derivovanie (derivácia súčtu, súčinu, podielu), derivácia inverznej a zloženej funkcie, logaritmická derivácia, diferenciál funkcie, aproximácia pomocou diferenciálu, absolútna a relatívna chyba, derivácie vyšších rádov, Leibnizov vzorec, derivácia funkcie zadanej parametricky, riešené príklady.

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Funkcie a ich vlastnosti

Vlastností funkcií, nutná podmienka existencie extrému, vety o strednej hodnote (Rolleho a Lagrengeova), L'Hospitalovo pravidlo a jeho použitie, Taylorov a Maclaurinov polynóm, aproximácia pomocou polynómu, riešené príklady.

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Vyšetrovanie priebehu funkcie

Vyšetrovanie priebehu funkcie, monotónnosť, postačujúca podmienka existencie lokálneho extrému, stacionárne body, lokálne a globálne extrémy, inflexné body, konvexnosť a konkávnosť, asymptotické vlastnosti, asymptoty bez smernice a  so smernicou, riešené príklady.

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Neurčitý integrál

Primitívna funkcia, neurčitý integrál, základné vlastnosti, integrály elementárnych funkcií, metóda rozkladu, metóda per partes, metóda neurčitých koeficientov, metóda substitúcie, integrály racionálnych funkcií, integrály iracionálnych funkcií, integrovanie goniometrických a hyperbolických funkcií.

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Neurčitý integrál

Riešené integrály:  $001.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sin{x}}.\enspace$ $002.\,{\displaystyle\int}\frac{\mathrm{d}x}{\cos{x}}.\enspace$ $003.\,{\displaystyle\int}\frac{\mathrm{d}x}{1+\sin{x}}.\enspace$ $004.\,{\displaystyle\int}\frac{\mathrm{d}x}{1+\cos{x}}.\enspace$ $005.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sinh{x}}.\enspace$ $006.\,{\displaystyle\int}\frac{\mathrm{d}x}{\cosh{x}}.\enspace$ $007.\,{\displaystyle\int}\frac{\mathrm{d}x}{1+\sinh{x}}.\enspace$ $008.\,{\displaystyle\int}\frac{\mathrm{d}x}{1+\cosh{x}}.\enspace$ $009.\,{\displaystyle\int}\sin^2{x}\,\mathrm{d}x.\enspace$ $010.\,{\displaystyle\int}\cos^2{x}\,\mathrm{d}x.\enspace$ $011.\,{\displaystyle\int}\sin^3{x}\,\mathrm{d}x.\enspace$ $012.\,{\displaystyle\int}\sin^{2n+1}{x}\,\mathrm{d}x.\enspace$ $013.\,{\displaystyle\int}\sin^n{x}\,\mathrm{d}x.\enspace$ $014.\,{\displaystyle\int}\cos^3{x}\,\mathrm{d}x.\enspace$ $015.\,{\displaystyle\int}\cos^{2n+1}{x}\,\mathrm{d}x.\enspace$ $016.\,{\displaystyle\int}\cos^n{x}\,\mathrm{d}x.\enspace$ $017.\,{\displaystyle\int}\sinh^2{x}\,\mathrm{d}x.\enspace$ $018.\,{\displaystyle\int}\cosh^2{x}\,\mathrm{d}x.\enspace$ $019.\,{\displaystyle\int}\sinh^3{x}\,\mathrm{d}x.\enspace$ $020.\,{\displaystyle\int}\sinh^{2n+1}{x}\,\mathrm{d}x.\enspace$ $021.\,{\displaystyle\int}\sinh^n{x}\,\mathrm{d}x.\enspace$ $022.\,{\displaystyle\int}\cosh^3{x}\,\mathrm{d}x.\enspace$ $023.\,{\displaystyle\int}\cosh^{2n+1}{x}\,\mathrm{d}x.\enspace$ $024.\,{\displaystyle\int}\cosh^n{x}\,\mathrm{d}x.\enspace$ $025.\,{\displaystyle\int}\mathrm{tg}^2{x}\,\mathrm{d}x.\enspace$ $026.\,{\displaystyle\int}\mathrm{tg}^3{x}\,\mathrm{d}x.\enspace$ $027.\,{\displaystyle\int}\mathrm{cotg}^2{x}\,\mathrm{d}x.\enspace$ $028.\,{\displaystyle\int}\mathrm{cotg}^3{x}\,\mathrm{d}x.\enspace$ $029.\,{\displaystyle\int}\mathrm{tgh}^2{x}\,\mathrm{d}x.\enspace$ $030.\,{\displaystyle\int}\mathrm{cotgh}^2{x}\,\mathrm{d}x.\enspace$ $031.\,{\displaystyle\int}\frac{\cos{x}}{4+3\sin{x}}\,\mathrm{d}x.\enspace$ $032.\,{\displaystyle\int}\frac{1+3\sin{x}+2\cos{x}}{\sin{2x}}\,\mathrm{d}x.\enspace$ $033.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sin^2{x}\cos^2{x}}.\enspace$ $034.\,{\displaystyle\int}\left[\mathrm{tg}\,{x}+\mathrm{cotg}\,{x}\right]\,\mathrm{d}x.\enspace$ $035.\,{\displaystyle\int}\frac{\mathrm{d}x}{a^2\cos^2{x}+b^2\sin^2{x}}.\enspace$ $036.\,{\displaystyle\int}\frac{\mathrm{d}x}{a^2\cos^2{x}-b^2\sin^2{x}}.\enspace$ $037.\,{\displaystyle\int}\frac{\mathrm{d}x}{4\cos^2{x}+\sin^2{x}}.\enspace$ $038.\,{\displaystyle\int}\frac{\mathrm{d}x}{4\cos^2{x}-\sin^2{x}}.\enspace$ $039.\,{\displaystyle\int}\frac{x^2\,\mathrm{d}x}{\sin{x^3}}.\enspace$ $040.\,{\displaystyle\int}x^2\sin{x^3}\,\mathrm{d}x.\enspace$ $041.\,{\displaystyle\int}\frac{\cos{x}\,\mathrm{d}x}{\sqrt[3]{\sin^2{x}}}.\enspace$ $042.\,{\displaystyle\int}\frac{\sin{x}\,\mathrm{d}x}{\sqrt{\cos^5{x}}}.\enspace$ $043.\,{\displaystyle\int}\frac{\cos{x}-\sin{x}}{\cos{x}+\sin{x}}\,\mathrm{d}x.\enspace$ $044.\,{\displaystyle\int}\frac{\ln{\cos{x}}}{\sin^2{x}}\,\mathrm{d}x.\enspace$ $045.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sin{x}\cos{x}}.\enspace$ $046.\,{\displaystyle\int}\frac{\mathrm{d}x}{\cos{x}+\sin{x}}.\enspace$ $047.\,{\displaystyle\int}\frac{(\sin{x}-\cos{x})\,\mathrm{d}x}{\sqrt[4]{\sin{x}+\cos{x}}}.\enspace$ $048.\,{\displaystyle\int}\frac{1-\mathrm{tg}\,{x}}{1+\mathrm{tg}\,{x}}\,\mathrm{d}x.\enspace$ $049.\,{\displaystyle\int}\cos^n{ax}\cdot\sin{ax}\,\mathrm{d}x.\enspace$ $050.\,{\displaystyle\int}\frac{\sin{ax}\,\mathrm{d}x}{\cos^n{ax}}.\enspace$ $051.\,{\displaystyle\int}\sin^n{ax}\cdot\cos{ax}\,\mathrm{d}x.\enspace$ $052.\,{\displaystyle\int}\frac{\cos{ax}\,\mathrm{d}x}{\sin^n{ax}}.\enspace$ $053.\,{\displaystyle\int}\sin{ax}\cdot\cos{bx}\,\mathrm{d}x.\enspace$ $054.\,{\displaystyle\int}\cos{ax}\cdot\cos{bx}\,\mathrm{d}x.\enspace$ $055.\,{\displaystyle\int}\sin{ax}\cdot\sin{bx}\,\mathrm{d}x.\enspace$ $056.\,{\displaystyle\int}\sin{ax}\cdot\cos{ax}\,\mathrm{d}x.\enspace$ $057.\,{\displaystyle\int}\cos^2{ax}\,\mathrm{d}x.\enspace$ $058.\,{\displaystyle\int}\sin^2{ax}\,\mathrm{d}x.\enspace$ $059.\,{\displaystyle\int}x\,\mathrm{tg}^2{x}\,\mathrm{d}x.\enspace$ $060.\,{\displaystyle\int}x\,\mathrm{cotg}^2{x}\,\mathrm{d}x.\enspace$ $061.\,{\displaystyle\int}x\sin{ax}\,\mathrm{d}x.\enspace$ $062.\,{\displaystyle\int}x\cos{ax}\,\mathrm{d}x.\enspace$ $063.\,{\displaystyle\int}x^2\sinh{ax}\,\mathrm{d}x.\enspace$ $064.\,{\displaystyle\int}x^2\sin{ax}\,\mathrm{d}x.\enspace$ $065.\,{\displaystyle\int}x^2\cosh{ax}\,\mathrm{d}x.\enspace$ $066.\,{\displaystyle\int}x^2\cos{ax}\,\mathrm{d}x.\enspace$ $067.\,{\displaystyle\int}x^n\sin{ax}\,\mathrm{d}x.\enspace$ $068.\,{\displaystyle\int}x^n\cos{ax}\,\mathrm{d}x.\enspace$ $069.\,{\displaystyle\int}x^3\sin{ax}\,\mathrm{d}x.\enspace$ $070.\,{\displaystyle\int}x^3\cos{ax}\,\mathrm{d}x.\enspace$ $071.\,{\displaystyle\int}\sqrt{1+\frac{1}{\sin{x}}}\,\mathrm{d}x.\enspace$ $072.\,{\displaystyle\int}\mathrm{arctg}\,{x}\,\mathrm{d}x.\enspace$ $073.\,{\displaystyle\int}x\,\mathrm{arctg}\,{x}\,\mathrm{d}x.\enspace$ $074.\,{\displaystyle\int}\ln{x}\,\mathrm{d}x.\enspace$ $075.\,{\displaystyle\int}\frac{\ln{\mathrm{arctg}\,{x}}}{x^2+1}\,\mathrm{d}x.\enspace$ $076.\,{\displaystyle\int}\ln{(x-1)^5}\,\mathrm{d}x.\enspace$ $077.\,{\displaystyle\int}\frac{\ln{x}}{x}\,\mathrm{d}x.\enspace$ $078.\,{\displaystyle\int}\frac{\mathrm{d}x}{x\ln{x}}.\enspace$ $079.\,{\displaystyle\int}x^2\ln{\sqrt{1-x}}.\enspace$ $080.\,{\displaystyle\int}\frac{\ln{x}}{\sqrt{x}}\,\mathrm{d}x.\enspace$ $081.\,{\displaystyle\int}x\ln{x}\,\mathrm{d}x.\enspace$ $082.\,{\displaystyle\int}x^2\ln{x}\,\mathrm{d}x.\enspace$ $083.\,{\displaystyle\int}x^n\ln{x}\,\mathrm{d}x.\enspace$ $084.\,{\displaystyle\int}(x+1)^2\ln{(x-1)^5}\,\mathrm{d}x.\enspace$ $085.\,{\displaystyle\int}x^x(\ln{x}+1)\,\mathrm{d}x.\enspace$ $086.\,{\displaystyle\int}x\ln^2{x}\,\mathrm{d}x.\enspace$ $087.\,{\displaystyle\int}\ln{(x^2+1)}\,\mathrm{d}x.\enspace$ $088.\,{\displaystyle\int}\ln{(\sqrt{1+x}+\sqrt{1-x})}\,\mathrm{d}x.\enspace$ $089.\,{\displaystyle\int}\ln{\big(x+\sqrt{x^2+1}\big)}\,\mathrm{d}x.\enspace$ $090.\,{\displaystyle\int}x(x-a)(x-b)\,\mathrm{d}x.\enspace$ $091.\,{\displaystyle\int}|x|\,\mathrm{d}x.\enspace$ $092.\,{\displaystyle\int}\min\limits_{x\in(0,\infty)}\{1,\frac1x\}\,\mathrm{d}x.\enspace$ $093.\,{\displaystyle\int}\frac{\mathrm{d}x}{5+4\mathrm{e}^x}.\enspace$ $094.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{5+4\mathrm{e}^x}}.\enspace$ $095.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{\mathrm{e}^{2x}+\mathrm{e}^x+1}}.\enspace$ $096.\,{\displaystyle\int}\sqrt{\frac{1-\mathrm{e}^x}{1+\mathrm{e}^x}}\,\mathrm{d}x.\enspace$ $097.\,{\displaystyle\int}(x+1)\mathrm{e}^x\,\mathrm{d}x.\enspace$ $098.\,{\displaystyle\int}x^2\mathrm{e}^{ax}\,\mathrm{d}x.\enspace$ $099.\,{\displaystyle\int}x^8\mathrm{e}^{ax}\,\mathrm{d}x.\enspace$ $100.\,{\displaystyle\int}x^n\mathrm{e}^x\,\mathrm{d}x.$

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Neurčitý integrál

Riešené integrály:  $101.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^2+4x+5}.\enspace$ $102.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^2+4x+3}.\enspace$ $103.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^2-4x+6}.\enspace$ $104.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^2-4x+2}.\enspace$ $105.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^2+4x+4}.\enspace$ $106.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x^2+4x+4}}.\enspace$ $107.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x^2+4x+5}}.\enspace$ $108.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x^2+4x+3}}.\enspace$ $109.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x^2-4x+6}}.\enspace$ $110.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x^2-4x+2}}.\enspace$ $111.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{-x^2+4x-5}}.\enspace$ $112.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{-x^2+4x-3}}.\enspace$ $113.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x(x-1)}}.\enspace$ $114.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x(2-x)}}.\enspace$ $115.\,{\displaystyle\int}\sqrt{x^2+4x+5}\,\mathrm{d}x.\enspace$ $116.\,{\displaystyle\int}\sqrt{x^2+4x+3}\,\mathrm{d}x.\enspace$ $117.\,{\displaystyle\int}\sqrt{x^2-4x+6}\,\mathrm{d}x.\enspace$ $118.\,{\displaystyle\int}\sqrt{x^2-4x+2}\,\mathrm{d}x.\enspace$ $119.\,{\displaystyle\int}\sqrt{-x^2+4x-3}\,\mathrm{d}x.\enspace$ $120.\,{\displaystyle\int}\sqrt{x(2-x)}\,\mathrm{d}x.\enspace$ $121.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x^2+a^2)^n}.\enspace$ $122.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x^2+a^2)^2}.\enspace$ $123.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x^2+a^2)^3}.\enspace$ $124.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x^2+a^2)^4}.\enspace$ $125.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x^2-a^2)^n}.\enspace$ $126.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x^2-a^2)^2}.\enspace$ $127.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x^2-a^2)^3}.\enspace$ $128.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x^2-a^2)^4}.\enspace$ $129.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x^2+4x+5)^2}.\enspace$ $130.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x^2+4x+3)^2}.\enspace$ $131.\,{\displaystyle\int}\frac{x^2\,\mathrm{d}x}{(x^2+a^2)^2}.\enspace$ $132.\,{\displaystyle\int}\frac{x^2\,\mathrm{d}x}{(x^2-a^2)^2}.\enspace$ $133.\,{\displaystyle\int}\frac{x\,\mathrm{d}x}{x^2+a^2}.\enspace$ $134.\,{\displaystyle\int}\frac{x\,\mathrm{d}x}{(x^2+a^2)^n}.\enspace$ $135.\,{\displaystyle\int}\frac{x\,\mathrm{d}x}{x^2-a^2}.\enspace$ $136.\,{\displaystyle\int}\frac{x\,\mathrm{d}x}{(x^2-a^2)^n}.\enspace$ $137.\,{\displaystyle\int}\frac{x^n\,\mathrm{d}x}{x-1}.\enspace$ $138.\,{\displaystyle\int}\frac{x\,\mathrm{d}x}{x-1}.\enspace$ $139.\,{\displaystyle\int}\frac{x^2\,\mathrm{d}x}{x-1}.\enspace$ $140.\,{\displaystyle\int}\frac{x^9\,\mathrm{d}x}{x-1}.\enspace$ $141.\,{\displaystyle\int}\frac{\mathrm{d}x}{(1-x)x^2}.\enspace$ $142.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^6(1+x^2)}.\enspace$ $143.\,{\displaystyle\int}\frac{(x-2)^4\,\mathrm{d}x}{(x-1)^2}.\enspace$ $144.\,{\displaystyle\int}\frac{(x-1)^4\,\mathrm{d}x}{(x-2)^2}.\enspace$ $145.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^3-7x-6}.\enspace$ $146.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^3-2x^2-x+2}.\enspace$ $147.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^3-3x-2}.\enspace$ $148.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^3+x^2-x-1}.\enspace$ $149.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^3-3x^2+4x-2}.\enspace$ $150.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^3-3x^2+3x-1}.\enspace$ $151.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^3-2x-4}.\enspace$ $152.\,{\displaystyle\int}\frac{\mathrm{d}x}{x^6+1}.\enspace$ $153.\,{\displaystyle\int}\frac{x\,\mathrm{d}x}{x^6+1}.\enspace$ $154.\,{\displaystyle\int}\frac{x^2\,\mathrm{d}x}{x^6+1}.\enspace$ $155.\,{\displaystyle\int}\frac{x^3\,\mathrm{d}x}{x^6+1}.\enspace$ $156.\,{\displaystyle\int}\frac{x^4\,\mathrm{d}x}{x^6+1}.\enspace$ $157.\,{\displaystyle\int}\frac{x^5\,\mathrm{d}x}{x^6+1}.\enspace$ $158.\,{\displaystyle\int}\frac{x^6\,\mathrm{d}x}{x^6+1}.\enspace$ $159.\,{\displaystyle\int}\frac{\mathrm{d}x}{2^x+1}.\enspace$ $160.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{2^x+1}}.\enspace$ $161.\,{\displaystyle\int}\frac{(1+x)\,\mathrm{d}x}{\sqrt{1-x^2}}.\enspace$ $162.\,{\displaystyle\int}\sqrt{\frac{1+x}{1-x}}\,\mathrm{d}x.\enspace$ $163.\,{\displaystyle\int}\sqrt{\frac{1-x}{1+x}}\,\mathrm{d}x.\enspace$ $164.\,{\displaystyle\int}\sqrt{\frac{x+1}{x-1}}\,\mathrm{d}x.\enspace$ $165.\,{\displaystyle\int}\sqrt{\frac{x-1}{x+1}}\,\mathrm{d}x.\enspace$ $166.\,{\displaystyle\int}\sqrt{\big(\frac{1+x}{1-x}\big)^3}\,\mathrm{d}x.\enspace$ $167.\,{\displaystyle\int}\sqrt{\big(\frac{1-x}{1+x}\big)^3}\,\mathrm{d}x.\enspace$ $168.\,{\displaystyle\int}\sqrt{\big(\frac{x+1}{x-1}\big)^3}\,\mathrm{d}x.\enspace$ $169.\,{\displaystyle\int}\sqrt{\big(\frac{x-1}{x+1}\big)^3}\,\mathrm{d}x.\enspace$ $170.\,{\displaystyle\int}\frac{\sqrt{1-\sqrt{x}}}{\sqrt{1+\sqrt{x}}}\,\mathrm{d}x.\enspace$ $171.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x+1}+\sqrt[3]{x+1}}.\enspace$ $172.\,{\displaystyle\int}\arcsin{\sqrt{\frac{1+x}{1-x}}}\,\mathrm{d}x.\enspace$ $173.\,{\displaystyle\int}\arcsin{\sqrt{\frac{1-x}{1+x}}}\,\mathrm{d}x.\enspace$ $174.\,{\displaystyle\int}\arcsin{\sqrt{\frac{x+1}{x-1}}}\,\mathrm{d}x.\enspace$ $175.\,{\displaystyle\int}\arcsin{\sqrt{\frac{x-1}{x+1}}}\,\mathrm{d}x.\enspace$ $176.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x-3}+\sqrt{x-5}}.\enspace$ $177.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x-3}-\sqrt{x-5}}.\enspace$ $178.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt{x-3}+\sqrt{5-x}}.\enspace$ $179.\,{\displaystyle\int}\frac{1+\sqrt{1-x^2}}{1-\sqrt{1-x^2}}\,\mathrm{d}x.\enspace$ $180.\,{\displaystyle\int}\sqrt{\frac{x}{1-x\sqrt{x}}}\,\mathrm{d}x.\enspace$ $181.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt[3]{x}+\sqrt[4]{x}}.\enspace$ $182.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt[6]{x}+\sqrt[4]{x}}.\enspace$ $183.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt[3]{x}+\sqrt[5]{x}}.\enspace$ $184.\,{\displaystyle\int}\frac{\mathrm{d}x}{\sqrt[3]{x}+1}.\enspace$ $185.\,{\displaystyle\int}\Big[\sqrt{x^3}-\frac{1}{\sqrt{x}}\Big]\,\mathrm{d}x.\enspace$ $186.\,{\displaystyle\int}\frac{x-1}{(\sqrt{x}+\sqrt[3]{x^2})x}\,\mathrm{d}x.\enspace$ $187.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x-1)\sqrt{x-2}}.\enspace$ $188.\,{\displaystyle\int}\frac{\mathrm{d}x}{(x+1)\sqrt{1-x}}.\enspace$ $189.\,{\displaystyle\int}(x-1)\sqrt{x-2}\,\mathrm{d}x.\enspace$ $190.\,{\displaystyle\int}\frac{\sqrt{1-x}}{x+1}\,\mathrm{d}x.\enspace$ $191.\,{\displaystyle\int}\frac{\mathrm{d}x}{\left(x-\sqrt{x^2-1}\right)^2}.\enspace$ $192.\,{\displaystyle\int}\frac{x^5\,\mathrm{d}x}{\sqrt{x^2+1}}.\enspace$ $193.\,{\displaystyle\int}\frac{x^5\,\mathrm{d}x}{\sqrt{x^3+1}}.\enspace$ $194.\,{\displaystyle\int}\frac{x^5\,\mathrm{d}x}{\sqrt{x^2-1}}.\enspace$ $195.\,{\displaystyle\int}\frac{x^5\,\mathrm{d}x}{\sqrt{x^3-1}}.\enspace$ $196.\,{\displaystyle\int}\frac{x^5\,\mathrm{d}x}{\sqrt{1-x^2}}.\enspace$ $197.\,{\displaystyle\int}\frac{x^5\,\mathrm{d}x}{\sqrt{1-x^3}}.\enspace$ $198.\,{\displaystyle\int}\frac{\sqrt{1-x^2}}{x^2}\,\mathrm{d}x.\enspace$ $199.\,{\displaystyle\int}\frac{\mathrm{d}x}{x\sqrt{x^4+x^2+1}}.\enspace$ $200.\,{\displaystyle\int}\frac{\mathrm{d}x}{x\sqrt{x^6+x^3+1}}.$

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Riemannov určitý integrál

Integrálne súčty, Riemannov určitý integrál, základné vlastnosti, grafická interpretácia, aditívnosť, výpočet Riemannovho integrálu, neurčitý Riemannov integrál, Newton-Leibnizov vzorec, metóda per partes, metóda substitúcie, integrovanie párnych a nepárnych funkcií, integrovanie periodických funkcií.

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