Integration of Transcendental Functions

Integrals containing the functions $\ln \varphi$, $\arctg \varphi$, $\arcsin \varphi$, where $\varphi$ is some algebraic function of $x$, are often integrated by parts, setting $u=\ln \varphi$, $u=\arctg \varphi$, $u=\arcsin \varphi$. This is how the integral $\int \ln x\, dx$ is found.

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

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

Find the integral $\int \frac{\ln (x-1)}{(x+1)^{3}}\, dx$.

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

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

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

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

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

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

Find the integral $\int x^{6}\arctg x\, dx$.

Find the integral $\int x^{7}\arctg x\, dx$.

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

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

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

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

Integrals containing the product of a polynomial with one of the functions $e^{ax}$, $\cos ax$, $\sin bx$ are best evaluated using the formula for repeated integration by parts, which can be given the form

If $u$ is a polynomial of degree $m$, then the last term vanishes.

Prove the equalities, in which $f(x)$ is a polynomial of degree $m$:

Prove the equality

Prove the equality

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

Find the integral $\int x^{5}\cos x\, dx$.

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

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

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

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

Find the integral $\int x^{3}e^{5x}\, dx$.

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

The following integrals, containing exponential functions, are found by integration by parts, and also by simple substitutions. Find the integral $\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\, dx$.

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

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

Find the integral $\int x^{5}e^{-x^{2}}\, dx$.

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

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

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

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

The following integrals are most conveniently found by using the fact that the integration formulas for functions containing $e^{ax}$ remain valid for complex values of $a$. Separating the real and imaginary parts, we obtain equalities with real integrals. This is how the integral $\int e^{ax}\cos bx\, dx$ is found.

Find the integral $\int e^{ax}\sin bx\, dx$.

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

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

A number of the following integrals belong to the type $\int f(\cos x, \sin x)\, dx$, where $f(u,v)$ is a rational function. Here it is useful to keep in mind three rules: 1) if $f(\cos x, \sin x)$ is multiplied by $-1$ when the sign is changed before one of the quantities $\cos x$ or $\sin x$, then it is useful to denote the other of them by $t$; 2) if $f(\cos x, \sin x)$ does not change when the sign is changed simultaneously before $\cos x$ and $\sin x$, then it is useful to set $\tg x=t$ or $\ctg x=t$; 3) the substitution $\tg \frac{x}{2}=t$ in all cases turns an integral of the type under consideration into an integral of a rational fraction. Unfortunately, this substitution leads to longer computations than the previous ones, when the latter are applicable.

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

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

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

Find the integral $\int \frac{dx}{\sin^{2}x\cos^{4}x}$.

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

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

Find the integral $\int \tg^{6}x\, dx$.

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

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

Find the integral $\int \sin^{7}x\cos^{6}x\, dx$.

Find the integral $\int \frac{\cos^{5}x}{\sin^{3}x}\, dx$.

Find the integral $\int \frac{\cos^{3}x}{\sin^{5}x}\, dx$.

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

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

For the integrals $U_{m,n}=\int \sin^{m}x\cos^{n}x\, dx$ there exist reduction formulas that allow reducing the integrals to others in which the indices $m$ and $n$ are closer to zero. These formulas reduce to four basic ones:

Denoting by $U_{n}$ the integral $U_{0,n}$ and by $V_{m}$ the integral $U_{m,0}$, we obtain from this two more reduction formulas for the integrals $U_{n}=\int \cos^{n}x\, dx$ and $V_{m}=\int \sin^{m}x\, dx$:

Using the reduction formulas, find the integral $\int \sin^{4}x\cos^{2}x\, dx$.

Find the integral $\int \sin^{6}x\cos^{4}x\, dx$.

Find the integral $\int \sin^{2}x\cos^{6}x\, dx$.

Find the integral $\int \sin^{4}x\cos^{6}x\, dx$.

Find the integral $\int \sin^{6}x\, dx$.

Find the integral $\int \sin^{8}x\, dx$.

Find the integral $\int \frac{\cos^{8}x}{\sin^{2}x}\, dx$.

Find the integral $\int \frac{\sin^{10}x}{\cos^{4}x}\, dx$.

Below is a series of integrals of trigonometric functions found by various techniques. Find the integral $\int \frac{\sin x-\cos x}{\sin x+\cos x}\, dx$, by comparing the numerator with the derivative of the denominator.

Find the integral $\int \frac{3\cos x+7\sin x}{5\cos x+2\sin x}\, dx$, by writing the numerator in the form $Au+Bu'$, where $u=5\cos x+2\sin x$.

Find the integral $\int \frac{dx}{\sin x+\cos x}$, by reducing the denominator to logarithmic form.

Apply the same technique to the integral $\int \frac{dx}{a\sin x+b\cos x}$.

Find the integral $\int \sqrt{1+\sin x}\, dx$, using the fact that $1=\cos^{2}\frac{x}{2}+\sin^{2}\frac{x}{2}$.

Using the substitution $\tg x=t$, find the integral $\int \frac{dx}{\sqrt[4]{\sin^{3}x\cos^{5}x}}$.

Find the integral $\int \sqrt{\frac{\sin^{3}x}{\cos^{7}x}}\, dx$.

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

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

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

Find the integral $\int \frac{dx}{4+3\tg x}$.

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

Find the integral $\int \frac{1+\tg^{2}x}{(4+\tg^{2}x)\tg^{3}x}\, dx$.

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

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

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

Find the integral $\int \frac{dx}{a^{2}-b^{2}\sin^{2}x}$; $a^{2}\neq b^{2}$.

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

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

Find the integrals. Find the integral $\int \frac{\sin 2x\, dx}{\sin^{4}x+\cos^{4}x}$.

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

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

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

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

Find the integral $\int \frac{\cos^{2}x}{\sin 4x}\, dx$.

Find the integral $\int \frac{\cos^{4}x}{\sin 6x}\, dx$.

Find the integral $\int \frac{\cos 2x}{\sin^{4}x}\, dx$.

Find the integral $\int \frac{\cos 2x}{\sin^{6}x}\, dx$.

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

Find the integral $\int \frac{\cos 3x}{\sin^{5}x}\, dx$.

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

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

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

Remark. Here the substitution $\frac{\pi }{4}-x=t$ is useful.

Find the integral $\int \frac{dx}{a+b\cos x}$ for $a^{2}>b^{2}$ and for $a^{2}<b^{2}$.

The following integrals contain hyperbolic functions. In these examples it is convenient to express them through exponential functions. Find the integral $\int \frac{x+\sh x}{\ch x-\sh x}\, dx$.

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

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

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

Find the integral $\int (\ch x+\sh x)\sqrt{\ch x-\sh x}\, dx$.

In the following examples it is more convenient not to express the hyperbolic functions through exponentials. Find the integral $\int \frac{dx}{(1-\ch x)^{2}}$.

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

Find the integral $\int \frac{\ch^{3}x\, dx}{1-\th x}$.

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

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

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

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

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