Integration Table


Table Of Basic Integrals

Basic Forms

\begin{equation}
\int x^n dx = \frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq -1
\end{equation}

\begin{equation}
\int \frac{1}{x}dx = \ln |x|
\end{equation}

\begin{equation}
\int u dv = uv – \int v du
\end{equation}

\begin{equation}
\int \frac{1}{ax+b}dx = \frac{1}{a} \ln |ax + b|
\end{equation}

Integrals of Rational Functions

\begin{equation}
\int \frac{1}{(x+a)^2}dx = -\frac{1}{x+a}
\end{equation}

\begin{equation}
\int (x+a)^n dx = \frac{(x+a)^{n+1}}{n+1}, n\ne -1
\end{equation}

\begin{equation}
\int x(x+a)^n dx = \frac{(x+a)^{n+1} ( (n+1)x-a)}{(n+1)(n+2)}
\end{equation}

\begin{equation}
\int \frac{1}{1+x^2}dx = \tan^{-1}x
\end{equation}

\begin{equation}
\int \frac{1}{a^2+x^2}dx = \frac{1}{a}\tan^{-1}\frac{x}{a}
\end{equation}

\begin{equation}
\int \frac{x}{a^2+x^2}dx = \frac{1}{2}\ln|a^2+x^2|
\end{equation}

\begin{equation}
\int \frac{x^2}{a^2+x^2}dx = x-a\tan^{-1}\frac{x}{a}
\end{equation}

\begin{equation}
\int \frac{x^3}{a^2+x^2}dx = \frac{1}{2}x^2-\frac{1}{2}a^2\ln|a^2+x^2|
\end{equation}

\begin{equation}
\int \frac{1}{ax^2+bx+c}dx = \frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}
\end{equation}

\begin{equation}
\int \frac{1}{(x+a)(x+b)}dx = \frac{1}{b-a}\ln\frac{a+x}{b+x}, \text{ } a\ne b
\end{equation}

\begin{equation}
\int \frac{x}{(x+a)^2}dx = \frac{a}{a+x}+\ln |a+x|
\end{equation}

\begin{equation}
\int \frac{x}{ax^2+bx+c}dx = \frac{1}{2a}\ln|ax^2+bx+c|
-\frac{b}{a\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}
\end{equation}

Integrals with Roots

\begin{equation}
\int \sqrt{x-a}\ dx = \frac{2}{3}(x-a)^{3/2}
\end{equation}

\begin{equation}
\int \frac{1}{\sqrt{x\pm a}}\ dx = 2\sqrt{x\pm a}
\end{equation}

\begin{equation}\label{eq:Rigo}
\int \frac{1}{\sqrt{a-x}}\ dx = -2\sqrt{a-x}
\end{equation}

\begin{equation}\label{eq:Gilmore}
\int x\sqrt{x-a}\ dx =
\left\{
\begin{array}{l}
\frac{2 a}{3} \left({x-a}\right)^{3/2} +\frac{2 }{5}\left( {x-a}\right)^{5/2},\text{ or}
\\ \frac{2}{3} x(x-a)^{3/2} – \frac{4}{15} (x-a)^{5/2}, \text{ or}
\\ \frac{2}{15}(2a+3x)(x-a)^{3/2}
\end{array}
\right.
\end{equation}

\begin{equation}
\int \sqrt{ax+b}\ dx = \left(\frac{2b}{3a}+\frac{2x}{3}\right)\sqrt{ax+b}
\end{equation}

\begin{equation}
\int (ax+b)^{3/2}\ dx =\frac{2}{5a}(ax+b)^{5/2}
\end{equation}

\begin{equation}\label{eq:Weems}
\int \frac{x}{\sqrt{x\pm a} } \ dx = \frac{2}{3}(x\mp 2a)\sqrt{x\pm a}
\end{equation}

\begin{equation}
\int \sqrt{\frac{x}{a-x}}\ dx = -\sqrt{x(a-x)}
-a\tan^{-1}\frac{\sqrt{x(a-x)}}{x-a}
\end{equation}

\begin{equation}
\int \sqrt{\frac{x}{a+x}}\ dx = \sqrt{x(a+x)}
-a\ln \left [ \sqrt{x} + \sqrt{x+a}\right]
\end{equation}

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

\begin{equation}
\int \sqrt{x(ax+b)}\ dx = \frac{1}{4a^{3/2}}\left[(2ax + b)\sqrt{ax(ax+b)}
-b^2 \ln \left| a\sqrt{x} + \sqrt{a(ax+b)} \right| \right ]
\end{equation}

\begin{equation}
\int \sqrt{x^3(ax+b)} \ dx =\left [
\frac{b}{12a}-
\frac{b^2}{8a^2x}+
\frac{x}{3}\right]
\sqrt{x^3(ax+b)} +
\frac{b^3}{8a^{5/2}}\ln \left | a\sqrt{x} + \sqrt{a(ax+b)} \right |
\end{equation}

\begin{equation}
\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 | x + \sqrt{x^2\pm a^2} \right |
\end{equation}

\begin{equation}
\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}}
\end{equation}

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

\begin{equation}
\int \frac{1}{\sqrt{x^2 \pm a^2}}\ dx = \ln \left | x + \sqrt{x^2 \pm a^2} \right |
\end{equation}

\begin{equation}
\int \frac{1}{\sqrt{a^2 – x^2}}\ dx = \sin^{-1}\frac{x}{a}
\end{equation}

\begin{equation}
\int \frac{x}{\sqrt{x^2\pm a^2}}\ dx = \sqrt{x^2 \pm a^2}
\end{equation}

\begin{equation}
\int \frac{x}{\sqrt{a^2-x^2}}\ dx = -\sqrt{a^2-x^2}
\end{equation}

\begin{equation}\label{eq:Russ}
\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| x + \sqrt{x^2\pm a^2} \right |
\end{equation}

\begin{equation}\label{eq:Winokur1}
\int \sqrt{a x^2 + b x + c}\ dx =
\frac{b+2ax}{4a}\sqrt{ax^2+bx+c}
+
\frac{4ac-b^2}{8a^{3/2}}\ln \left| 2ax + b + 2\sqrt{a(ax^2+bx^+c)}\right |
\end{equation}

\begin{equation}\label{eq:Larry-Morris}\begin{split}
\int &x \sqrt{a x^2 + bx + c}\ dx = \frac{1}{48a^{5/2}}\left (
2 \sqrt{a} \sqrt{ax^2+bx+c}
\right .
\left( – 3b^2 + 2 abx + 8 a(c+ax^2) \right)
\\ & \left.
+ 3(b^3-4abc)\ln \left|b + 2ax + 2\sqrt{a}\sqrt{ax^2+bx+c} \right| \right)
\end{split}
\end{equation}

\begin{equation}
\int\frac{1}{\sqrt{ax^2+bx+c}}\ dx=
\frac{1}{\sqrt{a}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right |
\end{equation}

\begin{equation}\label{eq:Duley}
\int \frac{x}{\sqrt{ax^2+bx+c}}\ dx=
\frac{1}{a}\sqrt{ax^2+bx + c}

\frac{b}{2a^{3/2}}\ln \left| 2ax+b + 2 \sqrt{a(ax^2+bx+c)} \right |
\end{equation}

\begin{equation}\label{eq:Winokur2}
\int\frac{dx}{(a^2+x^2)^{3/2}}=\frac{x}{a^2\sqrt{a^2+x^2}}
\end{equation}

Integrals with Logarithms

\begin{equation}
\int \ln ax\ dx = x \ln ax – x
\end{equation}

\begin{equation}
\int x \ln x \ dx = \frac{1}{2} x^2 \ln x-\frac{x^2}{4}
\end{equation}

\begin{equation}
\int x^2 \ln x \ dx = \frac{1}{3} x^3 \ln x-\frac{x^3}{9}
\end{equation}

\begin{equation}
\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
\end{equation}

\begin{equation}
\int \frac{\ln ax}{x}\ dx = \frac{1}{2}\left ( \ln ax \right)^2
\end{equation}

\begin{equation}
\int \frac{\ln x}{x^2}\ dx = -\frac{1}{x}-\frac{\ln x}{x}
\end{equation}

\begin{equation}
\int \ln (ax + b) \ dx = \left ( x + \frac{b}{a} \right) \ln (ax+b) – x , a\ne 0
\end{equation}

\begin{equation}
\int \ln ( x^2 + a^2 )\hspace{.5ex} {dx} = x \ln (x^2 + a^2 ) +2a\tan^{-1} \frac{x}{a} – 2x
\end{equation}

\begin{equation}
\int \ln ( x^2 – a^2 )\hspace{.5ex} {dx} = x \ln (x^2 – a^2 ) +a\ln \frac{x+a}{x-a} – 2x
\end{equation}

\begin{equation}
\int \ln \left ( ax^2 + bx + c\right) \ dx = \frac{1}{a}\sqrt{4ac-b^2}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}
-2x + \left( \frac{b}{2a}+x \right )\ln \left (ax^2+bx+c \right)
\end{equation}

\begin{equation}
\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)
\end{equation}

\begin{equation}
\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)
\end{equation}

\begin{equation}
\int (\ln x)^2\ dx = 2x – 2x \ln x + x (\ln x)^2
\end{equation}

\begin{equation}
\int (\ln x)^3\ dx = -6 x+x (\ln x)^3-3 x (\ln x)^2+6 x \ln x
\end{equation}

\begin{equation}
\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
\end{equation}

\begin{equation}
\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
\end{equation}

Integrals with Exponentials

\begin{equation}
\int e^{ax}\ dx = \frac{1}{a}e^{ax}
\end{equation}

\begin{equation}\label{eq:ajoy}
\int \sqrt{x} e^{ax}\ dx = \frac{1}{a}\sqrt{x}e^{ax}
+\frac{i\sqrt{\pi}}{2a^{3/2}}
\text{erf}\left(i\sqrt{ax}\right),
\text{ where erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt
\end{equation}

\begin{equation}
\int x e^x\ dx = (x-1) e^x
\end{equation}

\begin{equation}
\int x e^{ax}\ dx = \left(\frac{x}{a}-\frac{1}{a^2}\right) e^{ax}
\end{equation}

\begin{equation}
\int x^2 e^{x}\ dx = \left(x^2 – 2x + 2\right) e^{x}
\end{equation}

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

\begin{equation}
\int x^3 e^{x}\ dx = \left(x^3-3x^2 + 6x – 6\right) e^{x}
\end{equation}

\begin{equation}\label{eq:swift1}
\int x^n e^{ax}\ dx = \dfrac{x^n e^{ax}}{a} –
\dfrac{n}{a}\int x^{n-1}e^{ax}\hspace{1pt}\text{d}x
\end{equation}

\begin{equation}\label{eq:ebke}
\int x^n e^{ax}\ dx = \frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax],
\text{ where } \Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t}\hspace{2pt}\text{d}t
\end{equation}

\begin{equation}\label{eq:swift2}
\int e^{ax^2}\ dx = -\frac{i\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right)
\end{equation}

\begin{equation}\label{eq:swift3}
\int e^{-ax^2}\ dx = \frac{\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(x\sqrt{a}\right)
\end{equation}

\begin{equation}\label{eq:qarles1}
\int x e^{-ax^2}\ {dx} = -\dfrac{1}{2a}e^{-ax^2}
\end{equation}

\begin{equation}\label{eq:qarles2}
\int x^2 e^{-ax^2}\ {dx} = \dfrac{1}{4}\sqrt{\dfrac{\pi}{a^3}}\text{erf}(x\sqrt{a}) -\dfrac{x}{2a}e^{-ax^2}
\end{equation}

Integrals with Trigonometric Functions

\begin{equation}
\int \sin ax \ dx = -\frac{1}{a} \cos ax
\end{equation}

\begin{equation}
\int \sin^2 ax\ dx = \frac{x}{2} – \frac{\sin 2ax} {4a}
\end{equation}

\begin{equation}
\int \sin^3 ax \ dx = -\frac{3 \cos ax}{4a} + \frac{\cos 3ax} {12a}
\end{equation}

\begin{equation}
\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]
\end{equation}

\begin{equation}
\int \cos ax\ dx= \frac{1}{a} \sin ax
\end{equation}

\begin{equation}
\int \cos^2 ax\ dx = \frac{x}{2}+\frac{ \sin 2ax}{4a}
\end{equation}

\begin{equation}
\int \cos^3 ax dx = \frac{3 \sin ax}{4a}+\frac{ \sin 3ax}{12a}
\end{equation}

\begin{equation}
\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]
\end{equation}

\begin{equation}\label{eq:veky}
\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
\end{equation}

\begin{equation}
\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
\end{equation}

\begin{equation}
\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)}
\end{equation}

\begin{equation}
\int \sin^2 x \cos x\ dx = \frac{1}{3} \sin^3 x
\end{equation}

\begin{equation}
\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)}
\end{equation}

\begin{equation}
\int \cos^2 ax \sin ax\ dx = -\frac{1}{3a}\cos^3{ax}
\end{equation}

\begin{equation}
\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)}
\end{equation}

\begin{equation}
\int \sin^2 ax \cos^2 ax\ dx = \frac{x}{8}-\frac{\sin 4ax}{32a}
\end{equation}

\begin{equation}
\int \tan ax\ dx = -\frac{1}{a} \ln \cos ax
\end{equation}

\begin{equation}
\int \tan^2 ax\ dx = -x + \frac{1}{a} \tan ax
\end{equation}

\begin{equation}
\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)
\end{equation}

\begin{equation}
\int \tan^3 ax dx = \frac{1}{a} \ln \cos ax + \frac{1}{2a}\sec^2 ax
\end{equation}

\begin{equation}
\int \sec x \ dx = \ln | \sec x + \tan x | = 2 \tanh^{-1} \left (\tan \frac{x}{2} \right)
\end{equation}

\begin{equation}
\int \sec^2 ax\ dx = \frac{1}{a} \tan ax
\end{equation}

\begin{equation}\label{eq:Kloeppel}
\int \sec^3 x \ {dx} = \frac{1}{2} \sec x \tan x + \frac{1}{2}\ln | \sec x + \tan x |
\end{equation}

\begin{equation}
\int \sec x \tan x\ dx = \sec x
\end{equation}

\begin{equation}
\int \sec^2 x \tan x\ dx = \frac{1}{2} \sec^2 x
\end{equation}

\begin{equation}
\int \sec^n x \tan x \ dx = \frac{1}{n} \sec^n x , n\ne 0
\end{equation}

\begin{equation}
\int \csc x\ dx = \ln \left | \tan \frac{x}{2} \right| = \ln | \csc x – \cot x| + C
\end{equation}

\begin{equation}
\int \csc^2 ax\ dx = -\frac{1}{a} \cot ax
\end{equation}

\begin{equation}
\int \csc^3 x\ dx = -\frac{1}{2}\cot x \csc x + \frac{1}{2} \ln | \csc x – \cot x |
\end{equation}

\begin{equation}
\int \csc^nx \cot x\ dx = -\frac{1}{n}\csc^n x, n\ne 0
\end{equation}

\begin{equation}
\int \sec x \csc x \ dx = \ln | \tan x |
\end{equation}

Products of Trigonometric Functions and Monomials

\begin{equation}
\int x \cos x \ dx = \cos x + x \sin x
\end{equation}

\begin{equation}
\int x \cos ax \ dx = \frac{1}{a^2} \cos ax + \frac{x}{a} \sin ax
\end{equation}

\begin{equation}
\int x^2 \cos x \ dx = 2 x \cos x + \left ( x^2 – 2 \right ) \sin x
\end{equation}

\begin{equation}
\int x^2 \cos ax \ dx = \frac{2 x \cos ax }{a^2} + \frac{ a^2 x^2 – 2 }{a^3} \sin ax
\end{equation}

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

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

\begin{equation}
\int x \sin x\ dx = -x \cos x + \sin x
\end{equation}

\begin{equation}
\int x \sin ax\ dx = -\frac{x \cos ax}{a} + \frac{\sin ax}{a^2}
\end{equation}

\begin{equation}
\int x^2 \sin x\ dx = \left(2-x^2\right) \cos x + 2 x \sin x
\end{equation}

\begin{equation}
\int x^2 \sin ax\ dx =\frac{2-a^2x^2}{a^3}\cos ax +\frac{ 2 x \sin ax}{a^2}
\end{equation}

\begin{equation}\label{eq:xul}
\int x^n \sin x \ dx = -\frac{1}{2}(i)^n\left[ \Gamma(n+1, -ix)
– (-1)^n\Gamma(n+1, -ix)\right]
\end{equation}

\begin{equation}
\int x \cos^2 x \ dx = \frac{x^2}{4}+\frac{1}{8}\cos 2x + \frac{1}{4} x \sin 2x
\end{equation}

\begin{equation}
\int x \sin^2 x \ dx = \frac{x^2}{4}-\frac{1}{8}\cos 2x – \frac{1}{4} x \sin 2x
\end{equation}

\begin{equation}
\int x \tan^2 x \ dx = -\frac{x^2}{2} + \ln \cos x + x \tan x
\end{equation}

\begin{equation}
\int x \sec^2 x \ dx = \ln \cos x + x \tan x
\end{equation}

Products of Trigonometric Functions and Exponentials

\begin{equation}
\int e^x \sin x \ dx = \frac{1}{2}e^x (\sin x – \cos x)
\end{equation}

\begin{equation}\label{eq:ritzert}
\int e^{bx} \sin ax\ dx = \frac{1}{a^2+b^2}e^{bx} (b\sin ax – a\cos ax)
\end{equation}

\begin{equation}
\int e^x \cos x\ dx = \frac{1}{2}e^x (\sin x + \cos x)
\end{equation}

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

\begin{equation}
\int x e^x \sin x\ dx = \frac{1}{2}e^x (\cos x – x \cos x + x \sin x)
\end{equation}

\begin{equation}
\int x e^x \cos x\ dx = \frac{1}{2}e^x (x \cos x
– \sin x + x \sin x)
\end{equation}

Integrals of Hyperbolic Functions

\begin{equation}
\int \cosh ax\ dx =\frac{1}{a} \sinh ax
\end{equation}

\begin{equation}
\int e^{ax} \cosh bx \ dx =
\begin{cases}
\displaystyle{\frac{e^{ax}}{a^2-b^2} }[ a \cosh bx – b \sinh bx ] & a\ne b \\
\displaystyle{\frac{e^{2ax}}{4a} + \frac{x}{2}} & a = b
\end{cases}
\end{equation}

\begin{equation}
\int \sinh ax\ dx = \frac{1}{a} \cosh ax
\end{equation}

\begin{equation}
\int e^{ax} \sinh bx \ dx =
\begin{cases}
\displaystyle{\frac{e^{ax}}{a^2-b^2} }[ -b \cosh bx + a \sinh bx ] & a\ne b \\
\displaystyle{\frac{e^{2ax}}{4a} – \frac{x}{2}} & a = b
\end{cases}
\end{equation}

\begin{equation}\label{eq:yates}
\int \tanh ax\hspace{1.5pt} dx =\frac{1}{a} \ln \cosh ax
\end{equation}

\begin{equation}\label{eq:dewitt}
\int e^{ax} \tanh bx\ dx =
\begin{cases}
\displaystyle{ \frac{ e^{(a+2b)x}}{(a+2b)}
{_2F_1}\left[ 1+\frac{a}{2b},1,2+\frac{a}{2b}, -e^{2bx}\right] }& \\
\displaystyle{
\hspace{1cm}-\frac{1}{a}e^{ax}{_2F_1}\left[ 1, \frac{a}{2b},1+\frac{a}{2b}, -e^{2bx}\right]
}
& a\ne b \\
\displaystyle{\frac{e^{ax}-2\tan^{-1}[e^{ax}]}{a} } & a = b
\end{cases}
\end{equation}

\begin{equation}
\int \cos ax \cosh bx\ dx =
\frac{1}{a^2 + b^2} \left[
a \sin ax \cosh bx + b \cos ax \sinh bx
\right]
\end{equation}

\begin{equation}
\int \cos ax \sinh bx\ dx =
\frac{1}{a^2 + b^2} \left[
b \cos ax \cosh bx +
a \sin ax \sinh bx
\right]
\end{equation}

\begin{equation}
\int \sin ax \cosh bx \ dx =
\frac{1}{a^2 + b^2} \left[
-a \cos ax \cosh bx +
b \sin ax \sinh bx
\right]
\end{equation}

\begin{equation}
\int \sin ax \sinh bx \ dx =
\frac{1}{a^2 + b^2} \left[
b \cosh bx \sin ax –
a \cos ax \sinh bx
\right]
\end{equation}

\begin{equation}
\int \sinh ax \cosh ax dx=
\frac{1}{4a}\left[
-2ax + \sinh 2ax \right]
\end{equation}

\begin{equation}
\int \sinh ax \cosh bx \ dx =
\frac{1}{b^2-a^2}\left[
b \cosh bx \sinh ax
– a \cosh ax \sinh bx \right]
\end{equation}

Useful Integral Results For Mechanics

$$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2}} = \sqrt{\frac{\pi}{a}}$$

$$\int\limits^{+ \infty}_{- \infty}x^{2n} e^{-ax^{2}} = (-1)^{n} \frac{\partial^{n}}{\partial a^{n}}\sqrt{\frac{\pi}{a}}$$

$$\int\limits^{+ \infty}_{- \infty} e^{-ax^{2} + bx} = e^{\frac{b^2}{4a}}\sqrt{\frac{\pi}{a}}$$

$$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \sin^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 – \frac{6(-1)^n}{n^2 \pi^2} \right)$$

$$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x^{2} \cos^2 \left( \frac{n \pi x}{a} \right) = \frac{1}{24} a^{3} \left( 1 + \frac{6(-1)^n}{n^2 \pi^2} \right)$$

$$\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} x \cos \left( \frac{ \pi x}{a} \right) \sin \left( \frac{2 \pi x}{a} \right) = \frac{8a^2}{9 \pi ^2} $$

_ _ _ _ _ _ _ _ _ _ _

$$\int\limits^{a}_{b} \frac{dx}{\sqrt{\left(a-x \right) \left(x-b \right)}} = \pi \text{ for a > b}$$

$$\int\limits^{a}_{b} \frac{dx}{x\sqrt{\left(a-x \right) \left(x-b \right)}} = \frac{ \pi}{\sqrt{ab}} \text{ for a > b > 0}$$

$$\int\limits^{\frac{\pi}{2}}_{- \frac{\pi}{2}} \frac{dx}{1+ y \sin x} = \frac{\pi}{\sqrt{1 – y^2}} \text{ for -1 < y < 1}$$

 


 

Useful Integrals For Electromagnetism:

$$\int \frac{dx}{\sqrt{a^{2} – x^{2}}} = \text{arcsin} \, \frac{x}{a}$$

$$\int \frac{x dx}{\sqrt{a^{2} + x^{2}}} = \sqrt{a^{2} + x^{2}}$$

$$\int \frac{dx}{\sqrt{a^{2} +x^{2}}} = \text{ln} \, \left(x + \sqrt{a^{2} + x^{2}} \right)$$

$$\int \frac{dx}{a^{2} +x^{2}} = \frac{1}{a} \, \text{arctan} \, \frac{x}{a}$$

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

$$\int\frac{x \, dx}{ \left( a^{2}+x^{2} \right)^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{a^{2} + x^{2}}}$$

$$\int \frac{dx}{\sqrt{ (x – a)^{2} + b^{2}}} = \text{ln} \, \frac{1}{(a – x) + \sqrt{(a-x)^{2} + b^{2}}}$$

$$\int \frac{(x – a) \, dx}{\left[ (x-a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \, – \frac{1}{\sqrt{(x-a)^{2} + b^{2}}}$$

$$\int \frac{dx}{\left[ (x – a)^{2} + b^{2} \right]^{\frac{3}{2}}} = \frac{x – a}{b^{2} \sqrt{(x – a)^{2} +b^{2}}}$$

 

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