Basic Formulas for Integration
Below are some basic formulas that you may want to revise: \[ 1)\hspace{5pt} \int x^ndx = \frac{x^{n+1}}{n+1} + C, n \neq 1 \] \[ 2)\hspace{5pt} \int \cos{x}dx = \sin{x} + C \] \[ 3)\hspace{5pt} \int \sin{x}dx = -\cos{x} + C \] \[ 4)\hspace{5pt} \int \sec^2{x}dx = \tan{x} + C \] \[ 5)\hspace{5pt} \int cosec^2xdx = -\cot{x} + C \] \[ 6)\hspace{5pt} \int \sec{x}\tan{x}dx = \sec{x} + C \] \[ 7)\hspace{5pt} \int cosecxcotxdx = -cosecx + C \] \[ 8)\hspace{5pt} \int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1}{x} + C \] \[ 9)\hspace{5pt} \int \frac{dx}{\sqrt{1-x^2}} = -\cos^{-1}{x} + C \] \[ 10)\hspace{5pt} \int \frac{dx}{1+x^2} = tan^{-1}x + C \] \[ 11)\hspace{5pt} \int (3x+5)^{10}dx = \frac{(3x+5)^{11}}{(11)(3)} \]
Integration by parts
\[ \int uvdx \neq \int udx.\int vdx \]
\[ \int uvdx = u \int vdx - \int \left[\frac{d}{dx}(u) \int vdx \right]dx \]
Priority for types of different types of functions is given by ILATE:
I \(\longrightarrow \) Inverse Trigonometric function
L \(\longrightarrow \) Logarithmic function
A \(\longrightarrow \) Algebraic function
T \(\longrightarrow \) Trigonometric function
E \(\longrightarrow \) Exponential function
Example : \[ \int xe^xdx \] In this problem, our first function will be \(x\) and second function will be \(e^x\) \[ = x \int e^xdx - \int \left[\frac{d}{dx}(x) \int e^xdx \right]dx \] \[ = xe^x - \int e^x dx \] \[ = xe^x - e^x + C \]
