Integral Table (Revised Feb. 24, 1999)

Department of Mathematics and Statistics

I. Basic Functions - here $b \neq 0$

1. ${\displaystyle\int x^n \, dx = \frac1{n+1}x^{n+1}+C, \hspace{.2 cm}n \not=-1}$	2. ${\displaystyle\int \frac1x \, dx = \ln \vert x\vert +C }$
3. ${\displaystyle\int a^{bx} \, dx = \frac1{b \ln a} a^{bx} + C}$; in particular, ${\displaystyle\int e^{bx} \, dx = \frac1b e^{bx} +C}$
4. ${\displaystyle\int \sin(bx) \, dx = -\frac1b\cos(bx) + C}$	5. ${\displaystyle\int \cos(bx) \, dx = \frac1b\sin(bx) + C}$
6. ${\displaystyle\int \tan(bx) \, dx = \frac1b\ln\vert\sec(bx)\vert + C}$	7. ${\displaystyle\int \sec(bx) \, dx = \frac1b\ln\vert\tan(bx) +\sec(bx)\vert +C}$

II. Products of e^x, $\cos x$, $\sin x$

8.

${\displaystyle\int e^{ax} \sin(bx) \, dx = \frac1{a^2+b^2}e^{ax} \left[ a\sin (bx) - b\cos(bx) \right] +C }$

9.

${\displaystyle\int e^{ax} \cos(bx) \, dx = \frac1{a^2+b^2}e^{ax} \left[ a\cos(bx)+b\sin(bx) \right] +C }$

10.

${\displaystyle\int \sin(ax) \sin(bx) \, dx = \frac{\sin(a-b)x}{2(a-b)}- \frac{\sin(a+b)x}{2(a+b)}+C, \hspace{.2 cm}a^2 \not=b^2. }$

11.

${\displaystyle\int \cos(ax) \cos(bx) \, dx = \frac{\sin(a-b)x}{2(a-b)}+ \frac{\sin(a+b)x}{2(a+b)}+C, \hspace{.2 cm}a^2\not=b^2. }$

12.

${\displaystyle\int \sin(ax) \cos(bx) \, dx = -\frac{\cos(a-b)x}{2(a-b)}- \frac{\cos(a+b)x}{2(a+b)}+C, \hspace{.2 cm}a^2\not=b^2. }$

III. Integration by Parts

13. ${\displaystyle\int u \, dv = uv - \int v \, du}$; [in other notation: ${\displaystyle\int f(x) \, g(x) \, dx = f(x) \, {\textstyle \int \! \! g} - \int f'(x) \, {\textstyle \int \! \! g} \, dx }$]

IV. Products of xⁿ with $\ln x$, e^x, $\sin x$ and $\cos x$ - here $b \neq 0$

14.

${\displaystyle\int x^n \ln x \, dx = \frac1{n+1} x^{n+1} \ln x - \frac1{(n+1)^2}x^{n+1}+C, \hspace{.2 cm}n \not=-1, x>0}$

15.

${\displaystyle\int x^n e^{bx} \, dx = \frac1bx^ne^{bx}-\frac{n}b \int x^{n-1}e^{bx} \, dx}$

16.

${\displaystyle\int x^n \sin bx \, dx = -\frac{x^n}b \cos bx + \frac{n}b \int x^{n-1} \cos bx \, dx }$

17.

${\displaystyle\int x^n \cos bx \, dx = \frac{x^n}b \sin bx - \frac{n}b \int x^{n-1} \sin bx \, dx }$

V. Integer Powers of $\sin x$, $\cos x$ and $\sec x$

18.

${\displaystyle\int \sin^n x \, dx = - \frac1n \sin^{n-1}x \cos x + \frac{n-1}n ... ... \sin^{n-2}x \, dx, \hspace{.2 cm}\mbox{mostly used when $n$\space positive } }$

19.

${\displaystyle\int \cos^n x \, dx = \frac1n \cos^{n-1} x \sin x + \frac{n-1}n \int \cos^{n-2} x \, dx, \hspace{.2 cm}\mbox{mostly used when $n$ positive }}$

20.

${\displaystyle\int \sec^m x \, dx = \frac1{m-1} \tan x \sec^{m-2}x+ \frac{m-2}{... ...x, \hspace{.2 cm}m \not= 1, \hspace{.2 cm}\mbox{mainly used for $m >0$\space }}$

21.

${\displaystyle\int \sin^mx \cos^nx\, dx}$: If m is odd, let $u=\cos x$. If n is odd, let $u=\sin x$. If both m and n are even, convert all functions to $\sin x$ (or $\cos x$) and use $\sin^2 x = \frac12-\frac12 \cos 2x$ (or $\cos^2x=\frac12+\frac12 \cos 2x$); or use instead one of the reduction formulas 18, 19.

VI. Quadratic in Denominator

22.

${\displaystyle\int \frac{bx+c}{x^2+a^2} \, dx = \frac{b}2 \ln\vert x^2+a^2\vert+ \frac{c}a \arctan \frac{x}a +C, \hspace{.2 cm}a\not=0}$

23.

${\displaystyle\int \frac{cx+d}{(x-a)(x-b)} \, dx = \frac1{a-b} [(ac+d)\ln\vert x-a\vert-(bc+d)\ln\vert x-b\vert]+C, \hspace{.2 cm}a\not=b.}$

24.

${\displaystyle\int \frac{cx+d}{(x-a)^2} \, dx = c \ln\vert x-a\vert-\frac{ac+d}{x-a} +C}$

VII. Integrals Involving $\sqrt{a^2-b^2x^2}$, $\sqrt{b^2x^2+K}$, etc.; here b>0

25.

${\displaystyle\int \frac1{\sqrt{a^2-b^2x^2}} \, dx = \frac1b\arcsin \frac{bx}a +C,\;\; a>0}$

26.

${\displaystyle\int \frac1{\sqrt{b^2x^2 + K}} \, dx = \frac1b\ln\vert bx+ \sqrt{b^2x^2 + K}\vert+C}$, any K

27.

${\displaystyle\int \sqrt{b^2 x^2+K} \, dx = \frac12x\sqrt{b^2x^2+K} +\frac{K}{2b}\ln\vert bx+\sqrt{K+b^2x^2}\vert+C}$, any K

28.

${\displaystyle\int \sqrt{a^2-b^2x^2} \, dx = \frac12x\sqrt{a^2-b^2x^2}+\frac{a^2}{2b}\arcsin \frac{bx}a+C,\;\; a>0}$