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Definite Integrals of Rational Functions

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

Integrals of Trigonometric Functions

List of integrals involving trigonometric functions \( \int \sin x ~  dx = -\cos x \)   \( \int \cos x ~ dx = \sin x \)   \( \int \sin^2x ~ dx= \frac{x}{2}-\frac{1}{4}\sin(2x) \)   \( \int \cos^2x ~ dx = \frac{x}{2}+\frac{1}{4}\sin(2x) \)   \( \int \sin^3x ~ dx = \frac{1}{3}\cos^3x-\cos x \)   \( \int \cos^3x ~ dx […]

Integrals of Logarithmic Functions

List of integrals involving logarithmic functions \( \int \ln(cx)dx = x\ln(cx) – x \)   \( \int \ln(ax+b)dx = x\ln(ax+b) – x + \frac{b}{a}\ln(ax + b) \)   \( \int (\ln x)^2dx = x(\ln x)^2 – 2x\ln x + 2x \)   \( \int (\ln (cx))^ndx = x(\ln x)^n – n\cdot\int (\ln (cx))^{n-1}dx \)   \( \int \frac{dx}{\ln x} […]

Integrals of Exponential Functions

\( \int e^{cx}dx = \frac{1}{c}e^{cx} \)   \( \int a^{cx}dx = \frac{1}{c\cdot \ln a}a^{cx}, (\text{for } a>0, a\ne1 ) \)   \( \int x \cdot e^{cx} = \frac{e^{cx}}{c^2}(cx-1) \)   \( \int x^2 \cdot e^{cx} = e^{cx}\left(\frac{x^2}{c}-\frac{2x}{c^2} + \frac{2}{c^3}\right) \)   \( \int x^n \cdot e^{cx}dx = \frac{1}{c}x^ne^{cx}-\frac{n}{c}\int x^{n-1}e^{cx} dx \)   \( \int \frac{e^{cx}}{x} dx = \ln|x| + \sum\limits_{i=1}^\infty […]

Integrals of Rational Functions

Integrals involving \(ax + b\) \( \int (ax+b)^n dx = \frac{(ax+b)^{n+1}}{a(n+1)}, \quad (\text{for } n \ne 1) \) \( \int \frac{1}{ax+b}dx = \frac{1}{a}\ln|ax+b| \) \( \int x (ax+b)^ndx = \frac{a(n+1)x-b}{a^2(n+1)(n+2)}(ax+b)^{n+1}, \quad (\text{for } n \ne -1, n\ne-2) \) \( \int \frac{x}{ax+b}dx = \frac{x}{2} – \frac{b}{a^2}\ln|ax+b| \) \( \int \frac{x}{(ax+b)^2}dx = \frac{b}{a^2(ax+b)} – \frac{1}{a^2}\ln|ax+b| \) \( \int \frac{x^2}{ax+b} dx = \frac{1}{a^3} […]

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