Basic Formulas and Techniques of Integration

Applying formulas 1 and 2 and rules I and II, find the integral $\int x^{3} dx$.

Applying formulas 1 and 2 and rules I and II, find the integral $\int \sqrt{x}\, dx$.

Applying formulas 1 and 2 and rules I and II, find the integral $\int x \sqrt{x}\, dx$.

Applying formulas 1 and 2 and rules I and II, find the integral $\int \sqrt{x \sqrt{x \sqrt{x}}}\, dx$.

Find the integral $\int \frac{dx}{\sqrt{x}}$.

Find the integral $\int \frac{dx}{\sqrt[3]{x}}$.

Find the integral $\int \frac{dx}{x^{3}}$.

Find the integral $\int \frac{dx}{x \sqrt{x}}$.

Find the integral $\int (x^{2}+6x-5)\, dx$.

Find the integral $\int (x^{4}-3x^{2}+5x)\, dx$.

Find the integral $\int x^{2}(x^{2}-1)\, dx$.

Find the integral $\int (x^{2}-1)^{2}\, dx$.

Find the integral $\int \frac{x^{2}-3x+4}{x}\, dx$.

Find the integral $\int \frac{x^{3}+2x^{2}-x+3}{x}\, dx$.

Find the integral $\int \frac{x^{2}-x+1}{\sqrt{x}}\, dx$.

Find the integral $\int \frac{(x+1)^{2}}{\sqrt{x}}\, dx$.

The following integrals are found directly by applying formulas 1--12, and also by means of the change of variable $ax+b=t$. Find the integral $\int e^{-x}\, dx$.

Find the integral $\int e^{-3x}\, dx$.

Find the integral $\int \sin 2x\, dx$.

Find the integral $\int \cos (3x-5)\, dx$.

Find the integral $\int \frac{dx}{x-2}$.

Find the integral $\int \frac{dx}{3x+2}$.

Find the integral $\int \frac{x^{2}-x+1}{x-2}\, dx$.

Find the integral $\int \frac{dx}{x^{2}+2}$.

Find the integral $\int \frac{dx}{x^{3}-3}$.

Find the integral $\int \frac{dx}{2x^{2}+7}$.

Find the integral $\int \frac{dx}{\sqrt{5-x^{2}}}$.

Find the integral $\int \frac{x^{4}\, dx}{x-1}$.

Find the integral $\int \frac{dx}{\sqrt{x^{2}-6}}$.

Find the integral $\int \frac{dx}{\sqrt{x^{2}+7}}$.

Find the integral $\int \frac{dx}{(x-2)^{3}}$.

Find the integral $\int \frac{dx}{(2x-1)^{2}}$.

Find the integral $\int \frac{dx}{\cos^{2} 3x}$.

Find the integral $\int \frac{dx}{1+\cos x}$.

Find the integral $\int \frac{dx}{\sin^{2}(5x-2)}$.

Find the integral $\int \frac{dx}{\sqrt{2-3x}}$.

Find the integral $\int \frac{dx}{\sqrt{2-3x^{2}}}$.

Find the integral $\int \frac{dx}{\cos x}$.

Find the integral $\int 2^{x}\, dx$.

Find the integral $\int \frac{dx}{\sqrt{x-1}-\sqrt{x-2}}$, by rationalizing the numerator.

Find the integral $\int \frac{dx}{(x+2)^{2}+3}$.

Find the integral $\int \frac{dx}{x^{2}+6x+13}$.

Find the integral $\int \frac{dx}{x^{2}+5x+6}$.

Find the integral $\int \frac{dx}{\sqrt{x^{2}+2x+3}}$.

Find the integral $\int \frac{dx}{\sqrt{1+x-x^{2}}}$.

Find the integral $\int \frac{dx}{\sqrt{2x^{2}+3x}}$.

Find the integral $\int \frac{dx}{\sqrt{2ax-x^{2}}}$.

In a number of cases it is convenient to transform products and powers of sines and cosines into sums or differences. Find the integral $\int \cos x \cos 3x\, dx$.

Find the integral $\int \cos x \sin 3x\, dx$.

Find the integral $\int \cos 3x \cos 4x\, dx$.

Find the integral $\int \cos x \cos 3x \cos 5x\, dx$.

Find the integral $\int \sin x \sin 2x \sin 3x\, dx$.

Find the integral $\int \sin \frac{x}{2}\cos \frac{x}{12}\, dx$.

Find the integral $\int \sin^{2} x \cos^{2} 3x\, dx$.

Find the integral $\int \sin^{4} x\, dx$.

Find the integral $\int \cos^{4} x\, dx$.

Find the integral $\int \sin^{4} x \cos^{4} x\, dx$.

The following integrals are found by simple substitutions. Find the integral $\int \frac{e^{x}\, dx}{\sqrt{e^{x}+1}}$.

Find the integral $\int \frac{e^{x}\, dx}{e^{2x}+1}$.

Find the integral $\int \frac{\ln x}{x}\, dx$.

Find the integral $\int \frac{dx}{x \ln x}$.

Find the integral $\int \frac{\arctg x}{1+x^{2}}\, dx$.

Find the integral $\int \sin^{3}x\cos x\, dx$.

Find the integral $\int \cos^{2}x\sin x\, dx$.

Find the integral $\int e^{x^{2}}x\, dx$.

Find the integral $\int \frac{x\, dx}{(x^{2}+1)^{2}}$.

Find the integral $\int \frac{x\, dx}{(x^{2}+1)\sqrt{x^{2}+1}}$.

Find the integral $\int x^{2}\sqrt{x^{3}+1}\, dx$.

Find the integral $\int \frac{\cos x\, dx}{\sqrt[3]{\sin^{2}x}}$.

Find the integral $\int \frac{dx}{\arcsin x \sqrt{1-x^{2}}}$.

Find the integral $\int \frac{\sin x\, dx}{\sqrt{2+\cos x}}$.

Find the integral $\int \frac{\sin 2x\, dx}{\sin^{2}x+3}$.

Find the integral $\int \frac{\sqrt{\tg x}\, dx}{\cos^{2}x}$.

Find the integral $\int \frac{\cos^{3}x\, dx}{\sin^{4}x}$.

Find the integral $\int \frac{dx}{\sin^{2}x+5\cos^{2}x}$.

Find the integral $\int \tg x\, dx$.

Find the integral $\int \tg^{2}x\, dx$.

Find the integral $\int \frac{2x+11}{x^{2}+6x+13}\, dx$.

Find the integral $\int \frac{x\, dx}{x^{2}+x+1}$.

Find the integral $\int \frac{4x+8}{3x^{2}+2x+5}\, dx$.

Find the integral $\int \frac{5x+7}{\sqrt{5-4x-x^{2}}}\, dx$.

Find the integral $\int \frac{3x+2}{\sqrt{3+x+x^{2}}}\, dx$.

Find the integral $\int \frac{x\, dx}{\sqrt{x^{2}-ax}}$.

Find the integral $\int \frac{x\, dx}{\sqrt{x^{2}+x-1}}$.

Find the integral $\int \frac{x\, dx}{\sqrt{1+x-x^{2}}}$.

Find the integral $\int \frac{(x+a)\, dx}{\sqrt{ax-x^{2}}}$.

Using one of the trigonometric substitutions $x=a\sin \varphi$, $x=a\tg \varphi$, $x=a\sin^{2}\varphi$, find the integral $\int \frac{dx}{(a^{2}+x^{2})\sqrt{a^{2}+x^{2}}}$.

Find the integral $\int \frac{dx}{(1-x^{2})\sqrt{1-x^{2}}}$.

Find the integral $\int \frac{dx}{\sqrt{x(a-x)}}$.

Find the integral $\int \frac{x^{2}\, dx}{\sqrt{a^{2}-x^{2}}}$.

Find the integral $\int \sqrt{\frac{1-x}{1+x}}\, dx$, by rationalizing the denominator.

Applying integration by parts, expressed by equality III, find the integral $\int x^{2}\ln x\, dx$.

Find the integral $\int x\cos x\, dx$.

Find the integral $\int x\ch x\, dx$.

Find the integral $\int x\sin x\, dx$.

Find the integral $\int x\arctg x\, dx$.

Find the integral $\int (x^{2}+x)\ln (x+1)\, dx$.

Find the integral $\int x^{2}e^{-x}\, dx$.

Find the integral $\int x^{2}\sin x\, dx$.

Find the integral $\int x\ln (x^{2}-1)\, dx$.

Find the integral $\int \ln x\, dx$.

Find the integral $\int x\ln^{2}x\, dx$.