User:IntegralSam

Hey! I'm not very interesting ...

Integration Trick I've Accumulated!
I'll probably add more onto here sometime later, but I think this is a good start!

$$\int_{-\infty }^{\infty }e^{-ax^2+bx+c}dx=e^{c+\frac{b^2}{4a}}\sqrt{\frac{\pi}{a}}$$

$$\int_{0}^{\infty }\frac{ln(x)}{1+x^2}dx=0$$

$$\int_{0}^{\infty }\frac{1}{(1+x^2)(1+x^n)}dx=\frac{\pi}{4},n>0$$

$$\int_{0}^{1}\frac{x^n-1}{ln(x)}dx=ln(x+1),n>-1$$

$$\int_{0}^{\infty }f(x)dx=\frac{1}{2}\int_{0}^{\infty }\left ( f(x)+\frac{f(\frac{1}{x})}{x^2} \right )dx$$

$$\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$$

Here are more common tricks


 * Tangent half-angle substitution
 * Feynman's Integration Trick/ Differentiation Under the Integral Sign
 * Odd and even functions
 * $$\int_{-a}^{a }\frac{sin(x)}{1+x^{100}}dx=0$$ . The bounds are symmetrical, thus the magnitude of the areas will cancel each other out.