All Maths Formulas

Algebra

\int \infty 0 e - a x cos b x d x = a a 2 + b 2

$\int^\infty_0 e^{-ax} \cos bx \, dx = \frac{a}{a^2 + b^2}$

\int \infty 0 e - a x sin b x d x = b a 2 + b 2

$\int^\infty_0 e^{-ax} \sin bx \, dx = \frac{b}{a^2 + b^2}$

\int \infty 0 e - a x sin b x x d x = arctan b a

$\int^\infty_0 \frac{e^{-ax} \sin bx}{x} \, dx = \arctan \frac{b}{a}$

\int \infty 0 e - a x - e - b x x d x = ln b a

$\int^\infty_0 \frac{e^{-ax}-e^{-bx}}{x} dx = \ln \frac{b}{a}$

\int \infty 0 e - a x 2 d x = 1 2 π a - - \sqrt

$\int^\infty_0 e^{-ax^2} \, dx = \frac{1}{2} \sqrt{ \frac{\pi}{a} }$

\int \infty 0 e - a x 2 cos b x d x = 1 2 π a - - \sqrt e - b 2 4 a

$\int^\infty_0 e^{-ax^2} \cos bx \, dx = \frac{1}{2} \sqrt{ \frac{\pi}{a} } e^{-\frac{b^2}{4a}}$

\int \infty - \infty e - (a x 2 + b x + c) d x = π 2 - - \sqrt e b 2 - 4 a c 4 a

$\int^\infty_{-\infty} e^{-(ax^2+bx+c)} dx = \sqrt{\frac{\pi}{2}} e^\frac{b^2-4ac}{4a}$

\int \infty 0 x n e - a x d x = Γ ( n + 1 ) a n + 1

$\int^\infty_0 x^n\,e^{-ax}dx = \frac{\Gamma(n+1)}{a^{n+1}}$

\int \infty 0 x m e - a x 2 d x = Γ ( m + 1 2 ) 2 a ( m + 1 ) / 2

$\int^\infty_0 x^m\,e^{-ax^2}dx = \frac{\Gamma\left(\frac{m+1}{2}\right)}{2a^{(m+1)/2}}$

\int \infty 0 e - (a x 2 + b / x 2) d x = 1 2 π a - - \sqrt e - 2 a b \sqrt

$\int^\infty_0 e^{-\left(ax^2+b/x^2\right)} dx = \frac{1}{2}\sqrt{\frac{\pi}{a}}e^{-2\sqrt{ab}}$

\int \infty 0 x d x e x - 1 = π 2 6

$\int^\infty_0 \frac{x\,dx}{e^x-1} = \frac{\pi^2}{6}$

\int \infty 0 x n - 1 e x - 1 d x = Γ (n) (1 1 n + 1 2 n + 1 3 n + \dots)

$\int^\infty_0 \frac{x^{n-1}}{e^x-1}dx = \Gamma (n) \left( \frac{1}{1^n} + \frac{1}{2^n} + \frac{1}{3^n} + \cdots \right)$

\int \infty 0 x d x e x + 1 = π 2 12

$\int^\infty_0 \frac{x\,dx}{e^x+1} = \frac{\pi^2}{12}$

\int \infty 0 x n - 1 e x + 1 d x = Γ (n) (1 1 n - 1 2 n + 1 3 n - \dots)

$\int^\infty_0 \frac{x^{n-1}}{e^x+1}dx = \Gamma (n) \left( \frac{1}{1^n} - \frac{1}{2^n} + \frac{1}{3^n} - \cdots \right)$

\int \infty 0 sin m x e 2 π x - 1 d x = 1 4 coth m 2 - 1 2 m

$\int^\infty_0 \frac{\sin mx}{e^{2\pi x} - 1} dx = \frac{1}{4} \coth \frac{m}{2} - \frac{1}{2m}$

\int \infty 0 (1 1 + x - e - x) d x x = γ

$\int^\infty_0 \left( \frac{1}{1+x} - e^{-x} \right) \frac{dx}{x} = \gamma$

\int \infty 0 e - x 2 - e - x x d x = 1 2 γ

$\int^\infty_0 \frac{e^{-x^2} - e^{-x}}{x} dx = \frac{1}{2} \gamma$

\int \infty 0 (1 e x - 1 - e - x x) d x = γ

$\int^\infty_0 \left( \frac{1}{e^x-1} - \frac{e^{-x}}{x} \right) dx = \gamma$

\int \infty 0 e - a x - e - b x x sec ( p x ) d x = 1 2 ln (b 2 + p 2 a 2 + p 2)

$\int^\infty_0 \frac{e^{-ax} - e^{-bx}}{x \sec (px)} dx = \frac{1}{2} \ln\left( \frac{b^2+p^2}{a^2+p^2}\right)$

\int \infty 0 e - a x - e - b x x csc ( p x ) d x = arctan b p - arctan a p

$\int^\infty_0 \frac{e^{-ax} - e^{-bx}}{x \csc (px)} dx = \arctan \frac{b}{p} - \arctan \frac{a}{p}$

\int \infty 0 e - a x ( 1 - cos x ) x 2 d x = a r c c o t a - a 2 ln (a 2 + 1)

$\int^\infty_0 \frac{e^{-ax}(1-\cos x)}{x^2} dx = \mathrm{arccot}\,a - \frac{a}{2} \ln (a^2 + 1)$

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