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Definite Integrals of Logarithmic Functions

  \( \int^1_0 x^m(\ln x)^n dx = \frac{(-1)^n n!}{(m+1)^{n+1}} , \quad m>-1,\, n=0,1,2,\dots \)   \( \int^1_0 \frac{\ln x}{1+x} dx = -\frac{\pi^2}{12} \)   \( \int^1_0 \frac{\ln x}{1-x}dx = -\frac{\pi^2}{6} \)   \( \int^1_0 \frac{\ln (1+x)}{x} dx = \frac{\pi^2}{12} \)   \( \int^1_0 \frac{\ln (1 – x)}{x} dx = – \frac{\pi^2}{6} \)   \( \int^1_0 \ln x \ln (1+x) \,dx […]

Definite Integrals of Exponential Functions

\( \int^\infty_0 e^{-ax} \cos bx \, dx = \frac{a}{a^2 + b^2} \)   \( \int^\infty_0 e^{-ax} \sin bx \, dx = \frac{b}{a^2 + b^2} \)   \( \int^\infty_0 \frac{e^{-ax} \sin bx}{x} \, dx = \arctan \frac{b}{a} \)   \( \int^\infty_0 \frac{e^{-ax}-e^{-bx}}{x} dx = \ln \frac{b}{a} \)   \( \int^\infty_0 e^{-ax^2} \, dx = \frac{1}{2} \sqrt{ \frac{\pi}{a} } \)   \( […]

Definite Integrals of Trigonometric Functions

Basic formulas \( \int^{\pi/2}_0 \sin^2x\,dx = \int^{\pi/2}_0 \cos^2x\,dx = \frac{\pi}{4} \)   \( \int^\infty_0 \frac{\sin(px)}{x} \,dx = \left\{ \begin{array}{l l l} \pi/2 & p > 0 \\ ~0 & p = 0 \\ -\pi/2 & p < 0 \\ \end{array} \right. \)   \( \int^\infty_0 \frac{\sin^2px}{x^2} = \frac{\pi\,p}{2} \)   \( \int^\infty_0 \frac{1 – \cos(px)}{x^2}dx = \frac{\pi\,p}{2} \) […]

Definite Integrals of Rational Functions

\( \int^\infty_0 \frac{dx}{x^2+a^2} = \frac{\pi}{2a} \)   \( \int^\infty_0 \frac{x^{p-1}\,dx}{1+x} = \frac{\pi}{\sin (p\pi)} , ~ 0 < p < 1 \)   \( \int^\infty_0 \frac{x^m}{x^n + a^n} = \frac{\pi a^{m + 1 -n}}{n\,\sin[(m+1)\pi/n]}, ~0 < m + 1 < n \)   \( \int^a_0 \frac{dx}{\sqrt{a^2 – x^2}} = \frac{\pi}{2} \)   \( \int^a_0 \sqrt{a^2 – x^2}\,dx = \frac{\pi\,a^2}{4} \)   […]