Integrals Involving $\cot ax$

$\displaystyle \int \cot ax\,dx = \frac{1}{a}\ln(\sin ax)$

$\displaystyle \int \cot^2 ax\,dx = -\frac{\cot ax}{a} - x$

$\displaystyle \int \cot^3 ax\,dx = -\frac{\cot^2 ax}{2a} - \frac{1}{a}\ln(\sin ax)$

$\displaystyle \int \cot^n ax\csc^2 ax\,dx = -\frac{\cot^{n+1} ax}{(n + 1)a}$

$\displaystyle \int \frac{\csc^2 ax}{\cot ax}\,dx = -\frac{1}{a}\ln(\cot ax)$

$\displaystyle \int \frac{dx}{\cot ax} = -\frac{1}{a}\ln(\cos ax)$

$\displaystyle \int x\cot ax\,dx = \frac{1}{a^2}\bigg\{ax - \frac{(ax)^3}{9} - \frac{(ax)^5}{225} - \cdots - \frac{2^{2n}B_n(ax)^{2n+1}}{(2n + 1)!} - \cdots\bigg\}$

$\displaystyle \int \frac{\cot ax}{x}\,dx = -\frac{1}{ax} - \frac{ax}{3} - \frac{(ax)^3}{135} - \cdots - \frac{2^{2n}B_n(ax)^{2n-1}}{(2n - 1)(2n)!} - \cdots$

Here $B_n$ are the Bernoulli numbers: $B_1 = \frac{1}{6}$, $B_2 = \frac{1}{30}$, $B_3 = \frac{1}{42}$, $B_4 = \frac{1}{30}$, $B_5 = \frac{5}{66}, \ldots$

$\displaystyle \int x\cot^2 ax\,dx = -\frac{x\cot ax}{a} + \frac{1}{a^2}\ln(\sin ax) - \frac{x^2}{2}$

$\displaystyle \int \frac{dx}{p + q\cot ax} = \frac{px}{p^2 + q^2} - \frac{q}{a(p^2 + q^2)}\,\allowbreak\ln(p\sin ax +\nobreak q\cos ax)$

$\displaystyle \int \cot^n ax\,dx = -\frac{\cot^{n-1} ax}{(n - 1)a} - \int \cot^{n-2} ax\,dx$

Integrals Involving $\sec ax$

$\displaystyle \int \sec ax\,dx = \frac{1}{a}\ln(\sec ax +\nobreak \tan ax)$

$\displaystyle = \frac{1}{a}\ln\tan\left(\frac{\pi}{4} + \frac{ax}{2}\right)$

$\displaystyle \int \sec^2 ax\,dx = \frac{\tan ax}{a}$

$\displaystyle \int \sec^3 ax\,dx = \frac{\sec ax\tan ax}{2a} + \frac{1}{2a}\ln(\sec ax +\nobreak \tan ax)$

$\displaystyle \int \sec^n ax\tan ax\,dx = \frac{\sec^n ax}{na}$

$\displaystyle \int \frac{dx}{\sec ax} = \frac{\sin ax}{a}$

$\displaystyle \int x\sec ax\,dx = \frac{1}{a^2}\bigg\{\frac{(ax)^2}{2} + \frac{(ax)^4}{8} + \frac{5(ax)^6}{144} + \cdots + \frac{E_n(ax)^{2n+2}}{(2n + 2)(2n)!} + \cdots\bigg\}$

$\displaystyle \int \frac{\sec ax}{x}\,dx = \ln x + \frac{(ax)^2}{4} + \frac{5(ax)^4}{96} + \frac{61(ax)^6}{4320} + \cdots + \frac{E_n(ax)^{2n}}{2n(2n)!} + \cdots$

Here $E_n$ are the Euler numbers: $E_0 = 1$, $E_1 = 1$, $E_2 = 5$, $E_3 = 61$, $E_4 = 1385, \ldots$

$\displaystyle \int x\sec^2 ax\,dx = \frac{x}{a}\tan ax + \frac{1}{a^2}\ln(\cos ax)$

$\displaystyle \int \frac{dx}{q + p\sec ax} = \frac{x}{q} - \frac{p}{q}\int \frac{dx}{p + q\cos ax}$

$\displaystyle \int \sec^n ax\,dx = \frac{\sec^{n-2} ax\tan ax}{a(n - 1)} + \frac{n - 2}{n - 1}\int \sec^{n-2} ax\,dx$

Integrals Involving $\csc ax$

$\displaystyle \int \csc ax\,dx = \frac{1}{a}\ln(\csc ax -\nobreak \cot ax)$

$\displaystyle = \frac{1}{a}\ln\tan\frac{ax}{2}$

$\displaystyle \int \csc^2 ax\,dx = -\frac{\cot ax}{a}$

$\displaystyle \int \csc^3 ax\,dx = -\frac{\csc ax\cot ax}{2a} + \frac{1}{2a}\ln\tan\frac{ax}{2}$

$\displaystyle \int \csc^n ax\cot ax\,dx = -\frac{\csc^n ax}{na}$

$\displaystyle \int \frac{dx}{\csc ax} = -\frac{\cos ax}{a}$

$\displaystyle \int x\csc ax\,dx = \frac{1}{a^2}\bigg\{ax + \frac{(ax)^3}{18} + \frac{7(ax)^5}{1800} + \cdots + \frac{2(2^{2n-1} - 1)B_n(ax)^{2n+1}}{(2n + 1)!} + \cdots\bigg\}$

$\displaystyle \int \frac{\csc ax}{x}\,dx = -\frac{1}{ax} + \frac{ax}{6} + \frac{7(ax)^3}{1080} + \cdots + \frac{2(2^{2n-1} - 1)B_n(ax)^{2n-1}}{(2n - 1)(2n)!} + \cdots$

$\displaystyle \int x\csc^2 ax\,dx = -\frac{x}{a}\cot ax + \frac{1}{a^2}\ln(\sin ax)$

$\displaystyle \int \frac{dx}{q + p\csc ax} = \frac{x}{q} - \frac{p}{q}\int \frac{dx}{p + q\sin ax}$

$\displaystyle \int \csc^n ax\,dx = -\frac{\csc^{n-2} ax\cot ax}{a(n - 1)} + \frac{n - 2}{n - 1}\int \csc^{n-2} ax\,dx$

Integrals Involving Inverse Trigonometric Functions

$\displaystyle \int \sin^{-1}\frac{x}{a}\,dx = x\sin^{-1}\frac{x}{a} + \sqrt{a^2 - x^2}$

$\displaystyle \int x\sin^{-1}\frac{x}{a}\,dx = \left(\frac{x^2}{2} - \frac{a^2}{4}\right)\sin^{-1}\frac{x}{a} + \frac{x\sqrt{a^2 - x^2}}{4}$

$\displaystyle \int x^2\sin^{-1}\frac{x}{a}\,dx = \frac{x^3}{3}\sin^{-1}\frac{x}{a} + \frac{(x^2 + 2a^2)\sqrt{a^2 - x^2}}{9}$

$\displaystyle \int \frac{\sin^{-1}(x/a)}{x}\,dx = \frac{x}{a} + \frac{(x/a)^3}{2\cdot 3\cdot 3} + \frac{1\cdot 3\,(x/a)^5}{2\cdot 4\cdot 5\cdot 5} + \frac{1\cdot 3\cdot 5\,(x/a)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7} + \cdots$

$\displaystyle \int \frac{\sin^{-1}(x/a)}{x^2}\,dx = -\frac{\sin^{-1}(x/a)}{x} - \frac{1}{a}\ln\left(\frac{a + \sqrt{a^2 - x^2}}{x}\right)$

$\displaystyle \int \left(\sin^{-1}\frac{x}{a}\right)^2 dx = x\left(\sin^{-1}\frac{x}{a}\right)^2 - 2x + 2\sqrt{a^2 - x^2}\,\sin^{-1}\frac{x}{a}$

$\displaystyle \int \cos^{-1}\frac{x}{a}\,dx = x\cos^{-1}\frac{x}{a} - \sqrt{a^2 - x^2}$

$\displaystyle \int x\cos^{-1}\frac{x}{a}\,dx = \left(\frac{x^2}{2} - \frac{a^2}{4}\right)\cos^{-1}\frac{x}{a} - \frac{x\sqrt{a^2 - x^2}}{4}$

$\displaystyle \int x^2\cos^{-1}\frac{x}{a}\,dx = \frac{x^3}{3}\cos^{-1}\frac{x}{a} - \frac{(x^2 + 2a^2)\sqrt{a^2 - x^2}}{9}$

$\displaystyle \int \frac{\cos^{-1}(x/a)}{x}\,dx = \frac{\pi}{2}\ln x - \int \frac{\sin^{-1}(x/a)}{x}\,dx$

$\displaystyle \int \frac{\cos^{-1}(x/a)}{x^2}\,dx = -\frac{\cos^{-1}(x/a)}{x} + \frac{1}{a}\ln\left(\frac{a + \sqrt{a^2 - x^2}}{x}\right)$

$\displaystyle \int \left(\cos^{-1}\frac{x}{a}\right)^2 dx = x\left(\cos^{-1}\frac{x}{a}\right)^2 - 2x - 2\sqrt{a^2 - x^2}\,\cos^{-1}\frac{x}{a}$

$\displaystyle \int \tan^{-1}\frac{x}{a}\,dx = x\tan^{-1}\frac{x}{a} - \frac{a}{2}\ln(x^2 +\nobreak a^2)$

$\displaystyle \int x\tan^{-1}\frac{x}{a}\,dx = \frac{1}{2}(x^2 +\nobreak a^2)\tan^{-1}\frac{x}{a} - \frac{ax}{2}$

$\displaystyle \int x^2\tan^{-1}\frac{x}{a}\,dx = \frac{x^3}{3}\tan^{-1}\frac{x}{a} - \frac{ax^2}{6} + \frac{a^3}{6}\ln(x^2 +\nobreak a^2)$

$\displaystyle \int \frac{\tan^{-1}(x/a)}{x}\,dx = \frac{x}{a} - \frac{(x/a)^3}{3^2} + \frac{(x/a)^5}{5^2} - \frac{(x/a)^7}{7^2} + \cdots$ $[|x| \lt a]$

$\displaystyle \int \frac{\tan^{-1}(x/a)}{x^2}\,dx = -\frac{1}{x}\tan^{-1}\frac{x}{a} - \frac{1}{2a}\ln\left(\frac{x^2 + a^2}{x^2}\right)$

$\displaystyle \int \cot^{-1}\frac{x}{a}\,dx = x\cot^{-1}\frac{x}{a} + \frac{a}{2}\ln(x^2 +\nobreak a^2)$

$\displaystyle \int x\cot^{-1}\frac{x}{a}\,dx = \frac{1}{2}(x^2 +\nobreak a^2)\cot^{-1}\frac{x}{a} + \frac{ax}{2}$

$\displaystyle \int x^2\cot^{-1}\frac{x}{a}\,dx = \frac{x^3}{3}\cot^{-1}\frac{x}{a} + \frac{ax^2}{6} - \frac{a^3}{6}\ln(x^2 +\nobreak a^2)$

$\displaystyle \int \frac{\cot^{-1}(x/a)}{x}\,dx = \frac{\pi}{2}\ln x - \int \frac{\tan^{-1}(x/a)}{x}\,dx$

$\displaystyle \int \frac{\cot^{-1}(x/a)}{x^2}\,dx = -\frac{1}{x}\cot^{-1}\frac{x}{a} + \frac{1}{2a}\ln\left(\frac{x^2 + a^2}{x^2}\right)$

$\displaystyle \int \sec^{-1}\frac{x}{a}\,dx = x\sec^{-1}\frac{x}{a} - a\ln\left|x + \sqrt{x^2 - a^2}\right|$ $[0 \lt \sec^{-1}\frac{x}{a} \lt \frac{\pi}{2}]$

$\displaystyle \int \sec^{-1}\frac{x}{a}\,dx = x\sec^{-1}\frac{x}{a} + a\ln\left|x + \sqrt{x^2 - a^2}\right|$ $[\frac{\pi}{2} \lt \sec^{-1}\frac{x}{a} \lt \pi]$

$\displaystyle \int x\sec^{-1}\frac{x}{a}\,dx = \frac{x^2}{2}\sec^{-1}\frac{x}{a} - \frac{a\sqrt{x^2 - a^2}}{2}$ $[0 \lt \sec^{-1}\frac{x}{a} \lt \frac{\pi}{2}]$

$\displaystyle \int x\sec^{-1}\frac{x}{a}\,dx = \frac{x^2}{2}\sec^{-1}\frac{x}{a} + \frac{a\sqrt{x^2 - a^2}}{2}$ $[\frac{\pi}{2} \lt \sec^{-1}\frac{x}{a} \lt \pi]$

$\displaystyle \int x^2\sec^{-1}\frac{x}{a}\,dx = \frac{x^3}{3}\sec^{-1}\frac{x}{a} - \frac{ax\sqrt{x^2 - a^2}}{6} - \frac{a^3}{6}\ln\left|x + \sqrt{x^2 - a^2}\right|$ $[0 \lt \sec^{-1}\frac{x}{a} \lt \frac{\pi}{2}]$

$\displaystyle \int x^2\sec^{-1}\frac{x}{a}\,dx = \frac{x^3}{3}\sec^{-1}\frac{x}{a} + \frac{ax\sqrt{x^2 - a^2}}{6} + \frac{a^3}{6}\ln\left|x + \sqrt{x^2 - a^2}\right|$ $[\frac{\pi}{2} \lt \sec^{-1}\frac{x}{a} \lt \pi]$

$\displaystyle \int \frac{\sec^{-1}(x/a)}{x}\,dx = \frac{\pi}{2}\ln x + \frac{a}{x} + \frac{(a/x)^3}{2\cdot 3\cdot 3} + \frac{1\cdot 3\,(a/x)^5}{2\cdot 4\cdot 5\cdot 5} + \frac{1\cdot 3\cdot 5\,(a/x)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7} + \cdots$

$\displaystyle \int \frac{\sec^{-1}(x/a)}{x^2}\,dx = -\frac{\sec^{-1}(x/a)}{x} + \frac{\sqrt{x^2 - a^2}}{ax}$ $[0 \lt \sec^{-1}\frac{x}{a} \lt \frac{\pi}{2}]$

$\displaystyle \int \frac{\sec^{-1}(x/a)}{x^2}\,dx = -\frac{\sec^{-1}(x/a)}{x} - \frac{\sqrt{x^2 - a^2}}{ax}$ $[\frac{\pi}{2} \lt \sec^{-1}\frac{x}{a} \lt \pi]$

$\displaystyle \int \csc^{-1}\frac{x}{a}\,dx = x\csc^{-1}\frac{x}{a} + a\ln\left|x + \sqrt{x^2 - a^2}\right|$ $[0 \lt \csc^{-1}\frac{x}{a} \lt \frac{\pi}{2}]$

$\displaystyle \int \csc^{-1}\frac{x}{a}\,dx = x\csc^{-1}\frac{x}{a} - a\ln\left|x + \sqrt{x^2 - a^2}\right|$ $[-\frac{\pi}{2} \lt \csc^{-1}\frac{x}{a} \lt 0]$

$\displaystyle \int x\csc^{-1}\frac{x}{a}\,dx = \frac{x^2}{2}\csc^{-1}\frac{x}{a} + \frac{a\sqrt{x^2 - a^2}}{2}$ $[0 \lt \csc^{-1}\frac{x}{a} \lt \frac{\pi}{2}]$

$\displaystyle \int x\csc^{-1}\frac{x}{a}\,dx = \frac{x^2}{2}\csc^{-1}\frac{x}{a} - \frac{a\sqrt{x^2 - a^2}}{2}$ $[-\frac{\pi}{2} \lt \csc^{-1}\frac{x}{a} \lt 0]$

$\displaystyle \int x^2\csc^{-1}\frac{x}{a}\,dx = \frac{x^3}{3}\csc^{-1}\frac{x}{a} + \frac{ax\sqrt{x^2 - a^2}}{6} + \frac{a^3}{6}\ln\left|x + \sqrt{x^2 - a^2}\right|$ $[0 \lt \csc^{-1}\frac{x}{a} \lt \frac{\pi}{2}]$

$\displaystyle \int x^2\csc^{-1}\frac{x}{a}\,dx = \frac{x^3}{3}\csc^{-1}\frac{x}{a} - \frac{ax\sqrt{x^2 - a^2}}{6} - \frac{a^3}{6}\ln\left|x + \sqrt{x^2 - a^2}\right|$ $[-\frac{\pi}{2} \lt \csc^{-1}\frac{x}{a} \lt 0]$

$\displaystyle \int \frac{\csc^{-1}(x/a)}{x}\,dx = -\left(\frac{a}{x} + \frac{(a/x)^3}{2\cdot 3\cdot 3} + \frac{1\cdot 3\,(a/x)^5}{2\cdot 4\cdot 5\cdot 5} + \frac{1\cdot 3\cdot 5\,(a/x)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7} + \cdots\right)$

$\displaystyle \int \frac{\csc^{-1}(x/a)}{x^2}\,dx = -\frac{\csc^{-1}(x/a)}{x} - \frac{\sqrt{x^2 - a^2}}{ax}$ $[0 \lt \csc^{-1}\frac{x}{a} \lt \frac{\pi}{2}]$

$\displaystyle \int \frac{\csc^{-1}(x/a)}{x^2}\,dx = -\frac{\csc^{-1}(x/a)}{x} + \frac{\sqrt{x^2 - a^2}}{ax}$ $[-\frac{\pi}{2} \lt \csc^{-1}\frac{x}{a} \lt 0]$

$\displaystyle \int x^m\sin^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\sin^{-1}\frac{x}{a} - \frac{1}{m + 1}\int \frac{x^{m+1}}{\sqrt{a^2 - x^2}}\,dx$

$\displaystyle \int x^m\cos^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\cos^{-1}\frac{x}{a} + \frac{1}{m + 1}\int \frac{x^{m+1}}{\sqrt{a^2 - x^2}}\,dx$

$\displaystyle \int x^m\tan^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\tan^{-1}\frac{x}{a} - \frac{a}{m + 1}\int \frac{x^{m+1}}{x^2 + a^2}\,dx$

$\displaystyle \int x^m\cot^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}}{m + 1}\cot^{-1}\frac{x}{a} + \frac{a}{m + 1}\int \frac{x^{m+1}}{x^2 + a^2}\,dx$

$\displaystyle \int x^m\sec^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}\sec^{-1}(x/a)}{m + 1} - \frac{a}{m + 1}\int \frac{x^m\,dx}{\sqrt{x^2 - a^2}}$ $[0 \lt \sec^{-1}\frac{x}{a} \lt \frac{\pi}{2}]$

$\displaystyle \int x^m\sec^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}\sec^{-1}(x/a)}{m + 1} + \frac{a}{m + 1}\int \frac{x^m\,dx}{\sqrt{x^2 - a^2}}$ $[\frac{\pi}{2} \lt \sec^{-1}\frac{x}{a} \lt \pi]$

$\displaystyle \int x^m\csc^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}\csc^{-1}(x/a)}{m + 1} + \frac{a}{m + 1}\int \frac{x^m\,dx}{\sqrt{x^2 - a^2}}$ $[0 \lt \csc^{-1}\frac{x}{a} \lt \frac{\pi}{2}]$

$\displaystyle \int x^m\csc^{-1}\frac{x}{a}\,dx = \frac{x^{m+1}\csc^{-1}(x/a)}{m + 1} - \frac{a}{m + 1}\int \frac{x^m\,dx}{\sqrt{x^2 - a^2}}$ $[-\frac{\pi}{2} \lt \csc^{-1}\frac{x}{a} \lt 0]$

Integrals Involving $e^{ax}$

$\displaystyle \int e^{ax}\,dx = \frac{e^{ax}}{a}$

$\displaystyle \int xe^{ax}\,dx = \frac{e^{ax}}{a}\left(x - \frac{1}{a}\right)$

$\displaystyle \int x^2e^{ax}\,dx = \frac{e^{ax}}{a}\left(x^2 - \frac{2x}{a} + \frac{2}{a^2}\right)$

$\displaystyle \int x^ne^{ax}\,dx = \frac{x^ne^{ax}}{a} - \frac{n}{a}\int x^{n-1}e^{ax}\,dx$

$\displaystyle = \frac{e^{ax}}{a}\left(x^n - \frac{nx^{n-1}}{a} + \frac{n(n - 1)x^{n-2}}{a^2} - \cdots + \frac{(-1)^nn!}{a^n}\right)$ $[n \in \mathbb{N}]$

$\displaystyle \int \frac{e^{ax}}{x}\,dx = \ln x + \frac{ax}{1\cdot 1!} + \frac{(ax)^2}{2\cdot 2!} + \frac{(ax)^3}{3\cdot 3!} + \cdots$

$\displaystyle \int \frac{e^{ax}}{x^n}\,dx = -\frac{e^{ax}}{(n - 1)x^{n-1}} + \frac{a}{n - 1}\int \frac{e^{ax}}{x^{n-1}}\,dx$

$\displaystyle \int \frac{dx}{p + qe^{ax}} = \frac{x}{p} - \frac{1}{ap}\ln(p +\nobreak qe^{ax})$

$\displaystyle \int \frac{dx}{(p + qe^{ax})^2} = \frac{x}{p^2} + \frac{1}{ap(p + qe^{ax})} - \frac{1}{ap^2}\ln(p +\nobreak qe^{ax})$

$\displaystyle \int \frac{dx}{pe^{ax} + qe^{-ax}} = \frac{1}{a\sqrt{pq}}\tan^{-1}\left(\sqrt{\frac{p}{q}}\,e^{ax}\right)$ $[pq > 0]$

$\displaystyle \int \frac{dx}{pe^{ax} + qe^{-ax}} = \frac{1}{2a\sqrt{-pq}}\,\allowbreak\ln\left(\frac{e^{ax} - \sqrt{-q/p}}{e^{ax} + \sqrt{-q/p}}\right)$ $[pq \lt 0]$

$\displaystyle \int e^{ax}\sin bx\,dx = \frac{e^{ax}(a\sin bx - b\cos bx)}{a^2 + b^2}$

$\displaystyle \int e^{ax}\cos bx\,dx = \frac{e^{ax}(a\cos bx + b\sin bx)}{a^2 + b^2}$

$\displaystyle \int xe^{ax}\sin bx\,dx = \frac{xe^{ax}(a\sin bx - b\cos bx)}{a^2 + b^2} - \frac{e^{ax}\big((a^2 - b^2)\sin bx - 2ab\cos bx\big)}{(a^2 + b^2)^2}$

$\displaystyle \int xe^{ax}\cos bx\,dx = \frac{xe^{ax}(a\cos bx + b\sin bx)}{a^2 + b^2} - \frac{e^{ax}\big((a^2 - b^2)\cos bx + 2ab\sin bx\big)}{(a^2 + b^2)^2}$

$\displaystyle \int e^{ax}\ln x\,dx = \frac{e^{ax}\ln x}{a} - \frac{1}{a}\int \frac{e^{ax}}{x}\,dx$

$\displaystyle \int e^{ax}\sin^n bx\,dx = \frac{e^{ax}\sin^{n-1} bx}{a^2 + n^2b^2}(a\sin bx -\nobreak nb\cos bx) + \frac{n(n - 1)b^2}{a^2 + n^2b^2}\int e^{ax}\sin^{n-2} bx\,dx$

$\displaystyle \int e^{ax}\cos^n bx\,dx = \frac{e^{ax}\cos^{n-1} bx}{a^2 + n^2b^2}(a\cos bx +\nobreak nb\sin bx) + \frac{n(n - 1)b^2}{a^2 + n^2b^2}\int e^{ax}\cos^{n-2} bx\,dx$

Integrals Involving $\ln x$

$\displaystyle \int \ln x\,dx = x\ln x - x$

$\displaystyle \int x\ln x\,dx = \frac{x^2}{2}\left(\ln x - \frac{1}{2}\right)$

$\displaystyle \int x^m\ln x\,dx = \frac{x^{m+1}}{m + 1}\left(\ln x - \frac{1}{m + 1}\right)$ $[m \ne -1]$

$\displaystyle \int \frac{\ln x}{x}\,dx = \frac{1}{2}\ln^2 x$

$\displaystyle \int \frac{\ln x}{x^2}\,dx = -\frac{\ln x}{x} - \frac{1}{x}$

$\displaystyle \int \ln^2 x\,dx = x\ln^2 x - 2x\ln x + 2x$

$\displaystyle \int \frac{\ln^n x}{x}\,dx = \frac{\ln^{n+1} x}{n + 1}$ $[n \ne -1]$

$\displaystyle \int \frac{dx}{x\ln x} = \ln(\ln x)$

$\displaystyle \int \frac{dx}{\ln x} = \ln(\ln x) + \ln x + \frac{\ln^2 x}{2\cdot 2!} + \frac{\ln^3 x}{3\cdot 3!} + \cdots$

$\displaystyle \int \frac{x^m\,dx}{\ln x} = \ln(\ln x) + (m +\nobreak 1)\ln x + \frac{(m + 1)^2\ln^2 x}{2\cdot 2!} + \frac{(m + 1)^3\ln^3 x}{3\cdot 3!} + \cdots$

$\displaystyle \int \ln^n x\,dx = x\ln^n x - n\int \ln^{n-1} x\,dx$

$\displaystyle \int x^m\ln^n x\,dx = \frac{x^{m+1}\ln^n x}{m + 1} - \frac{n}{m + 1}\int x^m\ln^{n-1} x\,dx$ [if $m = -1$, see $\int \frac{\ln^n x}{x}\,dx$]

$\displaystyle \int \ln(x^2 +\nobreak a^2)\,dx = x\ln(x^2 +\nobreak a^2) - 2x + 2a\tan^{-1}\frac{x}{a}$

$\displaystyle \int \ln(x^2 -\nobreak a^2)\,dx = x\ln(x^2 -\nobreak a^2) - 2x + a\ln\left(\frac{x + a}{x - a}\right)$

$\displaystyle \int x^m\ln(x^2 \pm\nobreak a^2)\,dx = \frac{x^{m+1}\ln(x^2 \pm a^2)}{m + 1} - \frac{2}{m + 1}\int \frac{x^{m+2}}{x^2 \pm a^2}\,dx$

Integrals Involving $\sinh ax$

$\displaystyle \int \sinh ax\,dx = \frac{\cosh ax}{a}$

$\displaystyle \int x\sinh ax\,dx = \frac{x\cosh ax}{a} - \frac{\sinh ax}{a^2}$

$\displaystyle \int x^2\sinh ax\,dx = \left(\frac{x^2}{a} + \frac{2}{a^3}\right)\cosh ax - \frac{2x\sinh ax}{a^2}$

$\displaystyle \int \frac{\sinh ax}{x}\,dx = ax + \frac{(ax)^3}{3\cdot 3!} + \frac{(ax)^5}{5\cdot 5!} + \cdots$

$\displaystyle \int \frac{\sinh ax}{x^2}\,dx = -\frac{\sinh ax}{x} + a\int \frac{\cosh ax}{x}\,dx$

$\displaystyle \int \frac{dx}{\sinh ax} = \frac{1}{a}\ln\tanh\frac{ax}{2}$

$\displaystyle \int \frac{x\,dx}{\sinh ax} = \frac{1}{a^2}\bigg\{ax - \frac{(ax)^3}{18} + \frac{7(ax)^5}{1800} - \cdots + \frac{2(-1)^n(2^{2n-1} - 1)B_n(ax)^{2n+1}}{(2n + 1)!} + \cdots\bigg\}$

$\displaystyle \int \sinh^2 ax\,dx = \frac{\sinh ax\cosh ax}{2a} - \frac{x}{2}$

$\displaystyle \int x\sinh^2 ax\,dx = \frac{x\sinh 2ax}{4a} - \frac{\cosh 2ax}{8a^2} - \frac{x^2}{4}$

$\displaystyle \int \frac{dx}{\sinh^2 ax} = -\frac{\coth ax}{a}$

$\displaystyle \int \sinh ax\sinh px\,dx = \frac{\sinh(a + p)x}{2(a + p)} - \frac{\sinh(a - p)x}{2(a - p)}$ [if $a = \pm p$, see $\int \sinh^2 ax\,dx$]

$\displaystyle \int \sinh ax\sin px\,dx = \frac{a\cosh ax\sin px - p\sinh ax\cos px}{a^2 + p^2}$

$\displaystyle \int \sinh ax\cos px\,dx = \frac{a\cosh ax\cos px + p\sinh ax\sin px}{a^2 + p^2}$

$\displaystyle \int \frac{dx}{p + q\sinh ax} = \frac{1}{a\sqrt{p^2 + q^2}}\,\allowbreak\ln\left(\frac{qe^{ax} + p - \sqrt{p^2 + q^2}}{qe^{ax} + p + \sqrt{p^2 + q^2}}\right)$

$\displaystyle \int \frac{dx}{(p + q\sinh ax)^2} = -\frac{q\cosh ax}{a(p^2 + q^2)(p + q\sinh ax)} + \frac{p}{p^2 + q^2}\int \frac{dx}{p + q\sinh ax}$

$\displaystyle \int \frac{dx}{p^2 + q^2\sinh^2 ax} = \frac{1}{ap\sqrt{q^2 - p^2}}\tan^{-1}\frac{\sqrt{q^2 - p^2}\tanh ax}{p}$ $[q^2 > p^2]$

$\displaystyle \int \frac{dx}{p^2 + q^2\sinh^2 ax} = \frac{1}{2ap\sqrt{p^2 - q^2}}\,\allowbreak\ln\left(\frac{p + \sqrt{p^2 - q^2}\tanh ax}{p - \sqrt{p^2 - q^2}\tanh ax}\right)$ $[p^2 > q^2]$

$\displaystyle \int \frac{dx}{p^2 - q^2\sinh^2 ax} = \frac{1}{2ap\sqrt{p^2 + q^2}}\,\allowbreak\ln\left(\frac{p + \sqrt{p^2 + q^2}\tanh ax}{p - \sqrt{p^2 + q^2}\tanh ax}\right)$

$\displaystyle \int x^m\sinh ax\,dx = \frac{x^m\cosh ax}{a} - \frac{m}{a}\int x^{m-1}\cosh ax\,dx$

$\displaystyle \int \sinh^n ax\,dx = \frac{\sinh^{n-1} ax\cosh ax}{an} - \frac{n - 1}{n}\int \sinh^{n-2} ax\,dx$

$\displaystyle \int \frac{\sinh ax}{x^n}\,dx = -\frac{\sinh ax}{(n - 1)x^{n-1}} + \frac{a}{n - 1}\int \frac{\cosh ax}{x^{n-1}}\,dx$

$\displaystyle \int \frac{dx}{\sinh^n ax} = -\frac{\cosh ax}{a(n - 1)\sinh^{n-1} ax} - \frac{n - 2}{n - 1}\int \frac{dx}{\sinh^{n-2} ax}$

$\displaystyle \int \frac{x\,dx}{\sinh^n ax} = -\frac{x\cosh ax}{a(n - 1)\sinh^{n-1} ax} - \frac{1}{a^2(n - 1)(n - 2)\sinh^{n-2} ax} - \frac{n - 2}{n - 1}\int \frac{x\,dx}{\sinh^{n-2} ax}$