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

The Elements of Calculus

Calculus can be very intimidating. Basic methods in calculus are not so complicated but methods that were developed by mathematicians like Newton. He got answers for very elementary questions anout equations and curves etc. Many of the concepts in calculus are pretty intuitive, and many people are very interested to study it. Limits Basic concept […]

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 […]

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