Table of Integrals

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Contents

• Basic Forms
• Integrals of Rational Functions
• Integrals with Roots
• Integrals with Logarithms
• Integrals with Exponentials
• Integrals with Trigonometric Functions
• Products of Trigonometric Functions and Monomials
• Products of Trigonometric Functions and Exponentials
• Integrals of Hyperbolic Functions
• If you find an error
• Usage and attribution
• Acknowledgements
• Web Site Statistics
• About this document ...

Basic Forms

$$\int x^n dx = \frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq -1$$ (1)

$$\int \frac{1}{x}dx = \ln \vert x\vert$$ (2)

$$\int u dv = uv - \int v du$$ (3)

$$\int \frac{1}{ax+b}dx = \frac{1}{a} \ln \vert ax + b\vert$$ (4)

Integrals of Rational Functions

$$\int \frac{1}{(x+a)^2}dx = -\frac{1}{x+a}$$ (5)

$$\int (x+a)^n dx = \frac{(x+a)^{n+1}}{n+1}, n\ne -1$$ (6)

$$\int x(x+a)^n dx = \frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)}$$ (7)

$$\int \frac{1}{1+x^2}dx = \tan^{-1}x$$ (8)

$$\int \frac{1}{a^2+x^2}dx = \frac{1}{a}\tan^{-1}\frac{x}{a}$$ (9)

$$\int \frac{x}{a^2+x^2}dx = \frac{1}{2}\ln\vert a^2+x^2\vert$$ (10)

$$\int \frac{x^2}{a^2+x^2}dx = x-a\tan^{-1}\frac{x}{a}$$ (11)

$$\int \frac{x^3}{a^2+x^2}dx = \frac{1}{2}x^2-\frac{1}{2}a^2\ln\vert a^2+x^2\vert$$ (12)

$$\int \frac{1}{ax^2+bx+c}dx = \frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}$$ (13)

$$\int \frac{1}{(x+a)(x+b)}dx = \frac{1}{b-a}\ln\frac{a+x}{b+x}, a\ne b$$ (14)

$$\int \frac{x}{(x+a)^2}dx = \frac{a}{a+x}+\ln \vert a+x\vert$$ (15)

$$\int \frac{x}{ax^2+bx+c}dx = \frac{1}{2a}\ln\vert ax^2+bx+c\vert -\frac{b}{a\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}$$ (16)

Integrals with Roots

$$\int \sqrt{x-a} dx = \frac{2}{3}(x-a)^{3/2}$$ (17)

$$\int \frac{1}{\sqrt{x\pm a}} dx = 2\sqrt{x\pm a}$$ (18)

$$\int \frac{1}{\sqrt{a-x}} dx = -2\sqrt{a-x}$$ (19)

$$\int x\sqrt{x-a} dx = \left\{ \begin{array}{l} \frac{2 a}{3} \le... ...} (x-a)^{5/2}, \text{ or} \frac{2}{15}(2a+3x)(x-a)^{3/2} \end{array} \right.$$ (20)

$$\int \sqrt{ax+b} dx = \left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b}$$ (21)

$$\int (ax+b)^{3/2} dx =\frac{2}{5a}(ax+b)^{5/2}$$ (22)

$$\int \frac{x}{\sqrt{x\pm a} } dx = \frac{2}{3}(x\mp 2a)\sqrt{x\pm a}$$ (23)

$$\int \sqrt{\frac{x}{a-x}} dx = -\sqrt{x(a-x)} -a\tan^{-1}\frac{\sqrt{x(a-x)}}{x-a}$$ (24)

$$\int \sqrt{\frac{x}{a+x}} dx = \sqrt{x(a+x)} -a\ln \left [ \sqrt{x} + \sqrt{x+a}\right]$$ (25)

$$\int x \sqrt{ax + b} dx = \frac{2}{15 a^2}(-2b^2+abx + 3 a^2 x^2) \sqrt{ax+b}$$ (26)

$$\int \sqrt{x(ax+b)} dx = \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)} -b^2 \ln \left\vert a\sqrt{x} + \sqrt{a(ax+b)} \right\vert \right ]$$ (27)

$$\int \sqrt{x^3(ax+b)} dx =\left [ \frac{b}{12a}- \frac{b^2}{8a^... ...} + \frac{b^3}{8a^{5/2}}\ln \left \vert a\sqrt{x} + \sqrt{a(ax+b)} \right \vert$$ (28)

$$\int\sqrt{x^2 \pm a^2} dx = \frac{1}{2}x\sqrt{x^2\pm a^2} \pm\frac{1}{2}a^2 \ln \left \vert x + \sqrt{x^2\pm a^2} \right \vert$$ (29)

$$\int \sqrt{a^2 - x^2} dx = \frac{1}{2} x \sqrt{a^2-x^2} +\frac{1}{2}a^2\tan^{-1}\frac{x}{\sqrt{a^2-x^2}}$$ (30)

$$\int x \sqrt{x^2 \pm a^2} dx= \frac{1}{3}\left ( x^2 \pm a^2 \right)^{3/2}$$ (31)

$$\int \frac{1}{\sqrt{x^2 \pm a^2}} dx = \ln \left \vert x + \sqrt{x^2 \pm a^2} \right \vert$$ (32)

$$\int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1}\frac{x}{a}$$ (33)

$$\int \frac{x}{\sqrt{x^2\pm a^2}} dx = \sqrt{x^2 \pm a^2}$$ (34)

$$\int \frac{x}{\sqrt{a^2-x^2}} dx = -\sqrt{a^2-x^2}$$ (35)

$$\int \frac{x^2}{\sqrt{x^2 \pm a^2}} dx = \frac{1}{2}x\sqrt{x^2 \pm a^2} \mp \frac{1}{2}a^2 \ln \left\vert x + \sqrt{x^2\pm a^2} \right \vert$$ (36)

$$\int \sqrt{a x^2 + b x + c} dx = \frac{b+2ax}{4a}\sqrt{ax^2+bx+c... ...ac{4ac-b^2}{8a^{3/2}}\ln \left\vert 2ax + b + 2\sqrt{a(ax^2+bx^+c)}\right \vert$$ (37)

$$\begin{split}\int &x \sqrt{a x^2 + bx + c} dx = \frac{1}{48a... ...2ax + 2\sqrt{a}\sqrt{ax^2+bx+c} \right\vert \right) \end{split}$$ (38)

$$\int\frac{1}{\sqrt{ax^2+bx+c}} dx= \frac{1}{\sqrt{a}}\ln \left\vert 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right \vert$$ (39)

$$\int \frac{x}{\sqrt{ax^2+bx+c}} dx= \frac{1}{a}\sqrt{ax^2+bx + c} - \frac{b}{2a^{3/2}}\ln \left\vert 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right \vert$$ (40)

$$\int\frac{dx}{(a^2+x^2)^{3/2}}=\frac{x}{a^2\sqrt{a^2+x^2}}$$ (41)

Integrals with Logarithms

$$\int \ln ax dx = x \ln ax - x$$ (42)

$$\int x \ln x dx = \frac{1}{2} x^2 \ln x-\frac{x^2}{4}$$ (43)

$$\int x^2 \ln x dx = \frac{1}{3} x^3 \ln x-\frac{x^3}{9}$$ (44)

$$\int x^n \ln x dx = x^{n+1}\left( \dfrac{\ln x}{n+1}-\dfrac{1}{(n+1)^2}\right),\hspace{2ex} n\neq -1$$ (45)

$$\int \frac{\ln ax}{x} dx = \frac{1}{2}\left ( \ln ax \right)^2$$ (46)

$$\int \frac{\ln x}{x^2} dx = -\frac{1}{x}-\frac{\ln x}{x}$$ (47)

$$\int \ln (ax + b) dx = \left ( x + \frac{b}{a} \right) \ln (ax+b) - x , a\ne 0$$ (48)

$$\int \ln ( x^2 + a^2 )\hspace{.5ex} {dx} = x \ln (x^2 + a^2 ) +2a\tan^{-1} \frac{x}{a} - 2x$$ (49)

$$\int \ln ( x^2 - a^2 )\hspace{.5ex} {dx} = x \ln (x^2 - a^2 ) +a\ln \frac{x+a}{x-a} - 2x$$ (50)

$$\int \ln \left ( ax^2 + bx + c\right) dx = \frac{1}{a}\sqrt{4ac... ...sqrt{4ac-b^2}} -2x + \left( \frac{b}{2a}+x \right )\ln \left (ax^2+bx+c \right)$$ (51)

$$\int x \ln (ax + b) dx = \frac{bx}{2a}-\frac{1}{4}x^2 +\frac{1}{2}\left(x^2-\frac{b^2}{a^2}\right)\ln (ax+b)$$ (52)

$$\int x \ln \left ( a^2 - b^2 x^2 \right ) dx = -\frac{1}{2}x^2+ \frac{1}{2}\left( x^2 - \frac{a^2}{b^2} \right ) \ln \left (a^2 -b^2 x^2 \right)$$ (53)

$$\int (\ln x)^2 dx = 2x - 2x \ln x + x (\ln x)^2$$ (54)

$$\int (\ln x)^3 dx = -6 x+x (\ln x)^3-3 x (\ln x)^2+6 x \ln x$$ (55)

$$\int x (\ln x)^2 dx = \frac{x^2}{4}+\frac{1}{2} x^2 (\ln x)^2-\frac{1}{2} x^2 \ln x$$ (56)

$$\int x^2 (\ln x)^2 dx = \frac{2 x^3}{27}+\frac{1}{3} x^3 (\ln x)^2-\frac{2}{9} x^3 \ln x$$ (57)

Integrals with Exponentials

$$\int e^{ax} dx = \frac{1}{a}e^{ax}$$ (58)

$$\int \sqrt{x} e^{ax} dx = \frac{1}{a}\sqrt{x}e^{ax} +\frac{i\sqrt{\pi}}{2a^{3/2}} \left(i\sqrt{ax}\right), (x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$$ (59)

$$\int x e^x dx = (x-1) e^x$$ (60)

$$\int x e^{ax} dx = \left(\frac{x}{a}-\frac{1}{a^2}\right) e^{ax}$$ (61)

$$\int x^2 e^{x} dx = \left(x^2 - 2x + 2\right) e^{x}$$ (62)

$$\int x^2 e^{ax} dx = \left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax}$$ (63)

$$\int x^3 e^{x} dx = \left(x^3-3x^2 + 6x - 6\right) e^{x}$$ (64)

$$\int x^n e^{ax} dx = \dfrac{x^n e^{ax}}{a} - \dfrac{n}{a}\int x^{n-1}e^{ax} x$$ (65)

$$\int x^n e^{ax} dx = \frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax], \Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t} t$$ (66)

$$\int e^{ax^2} dx = -\frac{i\sqrt{\pi}}{2\sqrt{a}} \left(ix\sqrt{a}\right)$$ (67)

$$\int e^{-ax^2} dx = \frac{\sqrt{\pi}}{2\sqrt{a}} \left(x\sqrt{a}\right)$$ (68)

$$\int x e^{-ax^2} {dx} = -\dfrac{1}{2a}e^{-ax^2}$$ (69)

$$\int x^2 e^{-ax^2} {dx} = \dfrac{1}{4}\sqrt{\dfrac{\pi}{a^3}} (x\sqrt{a}) -\dfrac{x}{2a}e^{-ax^2}$$ (70)

Integrals with Trigonometric Functions

$$\int \sin ax dx = -\frac{1}{a} \cos ax$$ (71)

$$\int \sin^2 ax dx = \frac{x}{2} - \frac{\sin 2ax} {4a}$$ (72)

$$\int \sin^3 ax dx = -\frac{3 \cos ax}{4a} + \frac{\cos 3ax} {12a}$$ (73)

$$\int \sin^n ax dx = -\frac{1}{a}{\cos ax} \hspace{2mm}{_2F_1}\left[ \frac{1}{2}, \frac{1-n}{2}, \frac{3}{2}, \cos^2 ax \right]$$ (74)

$$\int \cos ax dx= \frac{1}{a} \sin ax$$ (75)

$$\int \cos^2 ax dx = \frac{x}{2}+\frac{ \sin 2ax}{4a}$$ (76)

$$\int \cos^3 ax dx = \frac{3 \sin ax}{4a}+\frac{ \sin 3ax}{12a}$$ (77)

$$\int \cos^p ax dx = -\frac{1}{a(1+p)}{\cos^{1+p} ax} \times {_2F_1}\left[ \frac{1+p}{2}, \frac{1}{2}, \frac{3+p}{2}, \cos^2 ax \right]$$ (78)

$$\int \cos x \sin x dx = \frac{1}{2}\sin^2 x + c_1 = -\frac{1}{2} \cos^2x + c_2 = -\frac{1}{4} \cos 2x + c_3$$ (79)

$$\int \cos ax \sin bx dx = \frac{\cos[(a-b) x]}{2(a-b)} - \frac{\cos[(a+b)x]}{2(a+b)} , a\ne b$$ (80)

$$\int \sin^2 ax \cos bx dx = -\frac{\sin[(2a-b)x]}{4(2a-b)} + \frac{\sin bx}{2b} - \frac{\sin[(2a+b)x]}{4(2a+b)}$$ (81)

$$\int \sin^2 x \cos x dx = \frac{1}{3} \sin^3 x$$ (82)

$$\int \cos^2 ax \sin bx dx = \frac{\cos[(2a-b)x]}{4(2a-b)} - \frac{\cos bx}{2b} - \frac{\cos[(2a+b)x]}{4(2a+b)}$$ (83)

$$\int \cos^2 ax \sin ax dx = -\frac{1}{3a}\cos^3{ax}$$ (84)

$$\int \sin^2 ax \cos^2 bx dx = \frac{x}{4} -\frac{\sin 2ax}{8a}- \frac{\sin[2(a-b)x]}{16(a-b)} +\frac{\sin 2bx}{8b}- \frac{\sin[2(a+b)x]}{16(a+b)}$$ (85)

$$\int \sin^2 ax \cos^2 ax dx = \frac{x}{8}-\frac{\sin 4ax}{32a}$$ (86)

$$\int \tan ax dx = -\frac{1}{a} \ln \cos ax$$ (87)

$$\int \tan^2 ax dx = -x + \frac{1}{a} \tan ax$$ (88)

$$\int \tan^n ax dx = \frac{\tan^{n+1} ax }{a(1+n)} \times {_2}F_1\left( \frac{n+1}{2}, 1, \frac{n+3}{2}, -\tan^2 ax \right)$$ (89)

$$\int \tan^3 ax dx = \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax$$ (90)

$$\int \sec x dx = \ln \vert \sec x + \tan x \vert = 2 \tanh^{-1} \left (\tan \frac{x}{2} \right)$$ (91)

$$\int \sec^2 ax dx = \frac{1}{a} \tan ax$$ (92)

$$\int \sec^3 x {dx} = \frac{1}{2} \sec x \tan x + \frac{1}{2}\ln \vert \sec x + \tan x \vert$$ (93)

$$\int \sec x \tan x dx = \sec x$$ (94)

$$\int \sec^2 x \tan x dx = \frac{1}{2} \sec^2 x$$ (95)

$$\int \sec^n x \tan x dx = \frac{1}{n} \sec^n x , n\ne 0$$ (96)

$$\int \csc x dx = \ln \left \vert \tan \frac{x}{2} \right\vert = \ln \vert \csc x - \cot x\vert + C$$ (97)

$$\int \csc^2 ax dx = -\frac{1}{a} \cot ax$$ (98)

$$\int \csc^3 x dx = -\frac{1}{2}\cot x \csc x + \frac{1}{2} \ln \vert \csc x - \cot x \vert$$ (99)

$$\int \csc^nx \cot x dx = -\frac{1}{n}\csc^n x, n\ne 0$$ (100)

$$\int \sec x \csc x dx = \ln \vert \tan x \vert$$ (101)

Products of Trigonometric Functions and Monomials

$$\int x \cos x dx = \cos x + x \sin x$$ (102)

$$\int x \cos ax dx = \frac{1}{a^2} \cos ax + \frac{x}{a} \sin ax$$ (103)

$$\int x^2 \cos x dx = 2 x \cos x + \left ( x^2 - 2 \right ) \sin x$$ (104)

$$\int x^2 \cos ax dx = \frac{2 x \cos ax }{a^2} + \frac{ a^2 x^2 - 2 }{a^3} \sin ax$$ (105)

$$\int x^n \cos x dx = -\frac{1}{2}(i)^{n+1}\left [ \Gamma(n+1, -ix) + (-1)^n \Gamma(n+1, ix)\right]$$ (106)

$$\int x^n \cos ax dx = \frac{1}{2}(ia)^{1-n}\left [ (-1)^n \Gamma(n+1, -iax) -\Gamma(n+1, ixa)\right]$$ (107)

$$\int x \sin x dx = -x \cos x + \sin x$$ (108)

$$\int x \sin ax dx = -\frac{x \cos ax}{a} + \frac{\sin ax}{a^2}$$ (109)

$$\int x^2 \sin x dx = \left(2-x^2\right) \cos x + 2 x \sin x$$ (110)

$$\int x^2 \sin ax dx =\frac{2-a^2x^2}{a^3}\cos ax +\frac{ 2 x \sin ax}{a^2}$$ (111)

$$\int x^n \sin x dx = -\frac{1}{2}(i)^n\left[ \Gamma(n+1, -ix) - (-1)^n\Gamma(n+1, -ix)\right]$$ (112)

$$\int x \cos^2 x dx = \frac{x^2}{4}+\frac{1}{8}\cos 2x + \frac{1}{4} x \sin 2x$$ (113)

$$\int x \sin^2 x dx = \frac{x^2}{4}-\frac{1}{8}\cos 2x - \frac{1}{4} x \sin 2x$$ (114)

$$\int x \tan^2 x dx = -\frac{x^2}{2} + \ln \cos x + x \tan x$$ (115)

$$\int x \sec^2 x dx = \ln \cos x + x \tan x$$ (116)

Products of Trigonometric Functions and Exponentials

$$\int e^x \sin x dx = \frac{1}{2}e^x (\sin x - \cos x)$$ (117)

$$\int e^{bx} \sin ax dx = \frac{1}{a^2+b^2}e^{bx} (b\sin ax - a\cos ax)$$ (118)

$$\int e^x \cos x dx = \frac{1}{2}e^x (\sin x + \cos x)$$ (119)

$$\int e^{bx} \cos ax dx = \frac{1}{a^2 + b^2} e^{bx} ( a \sin ax + b \cos ax )$$ (120)

$$\int x e^x \sin x dx = \frac{1}{2}e^x (\cos x - x \cos x + x \sin x)$$ (121)

$$\int x e^x \cos x dx = \frac{1}{2}e^x (x \cos x - \sin x + x \sin x)$$ (122)

Integrals of Hyperbolic Functions

$$\int \cosh ax dx =\frac{1}{a} \sinh ax$$ (123)

$$\int e^{ax} \cosh bx dx = \begin{cases}\displaystyle{\frac{e^{a... ...& a\ne b \displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b \end{cases}$$ (124)

$$\int \sinh ax dx = \frac{1}{a} \cosh ax$$ (125)

$$\int e^{ax} \sinh bx dx = \begin{cases}\displaystyle{\frac{e^{a... ...& a\ne b \displaystyle{\frac{e^{2ax}}{4a} - \frac{x}{2}} & a = b \end{cases}$$ (126)

$$\int \tanh ax\hspace{1.5pt} dx =\frac{1}{a} \ln \cosh ax$$ (127)

$$\int e^{ax} \tanh bx dx = \begin{cases}\displaystyle{ \frac{ e^{... ...ne b \displaystyle{\frac{e^{ax}-2\tan^{-1}[e^{ax}]}{a} } & a = b \end{cases}$$ (128)

$$\int \cos ax \cosh bx dx = \frac{1}{a^2 + b^2} \left[ a \sin ax \cosh bx + b \cos ax \sinh bx \right]$$ (129)

$$\int \cos ax \sinh bx dx = \frac{1}{a^2 + b^2} \left[ b \cos ax \cosh bx + a \sin ax \sinh bx \right]$$ (130)

$$\int \sin ax \cosh bx dx = \frac{1}{a^2 + b^2} \left[ -a \cos ax \cosh bx + b \sin ax \sinh bx \right]$$ (131)

$$\int \sin ax \sinh bx dx = \frac{1}{a^2 + b^2} \left[ b \cosh bx \sin ax - a \cos ax \sinh bx \right]$$ (132)

$$\int \sinh ax \cosh ax dx= \frac{1}{4a}\left[ -2ax + \sinh 2ax \right]$$ (133)

$$\int \sinh ax \cosh bx dx = \frac{1}{b^2-a^2}\left[ b \cosh bx \sinh ax - a \cosh ax \sinh bx \right]$$ (134)

If you find an error

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Acknowledgements

The author is not in any way affiliated with Wolfram Research, Mathematica, or the Wolfram Integrater. I've just posted the link at the top of this page because I think their web site is really cool!

Thanks to the following for pointing out corrections, typos, or suggesting new formulas: LS Rigo (19); Stephen Gilmore (20); Bruce Weems (23); Stephen Russ (36); Justin Winokur (37), (41); Larry Morris and Tom Gatliffe (38); James Duley (40); Daniel Ajoy (59); Johannes Ebke (66); Jim Swift (65),(66), (67), (68); Kregg Quarles (69),(70); Vedran (Veky) Čačić (79); Peter Kloeppel (93); Phillipe (Xul) (112); Nicole Ritzert and Andrea Bajo (118); Jose Antonio Alvarez Loyo Yates (127); Corné de Witt (128).

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