Basic Formula for Integration Engineering Maths, Btech first year

Basic Formulas for Integration

Below are some basic formulas that you may want to revise: $$1)\int x^{n}dx=\frac{x^{n+1}}{n+1}+C,n\ne 1$$ $$2)\int \cos xdx=\sin x+C$$ $$3)\int \sin xdx=-\cos x+C$$ $$4)\int {\sec }^{2}xdx=\tan x+C$$ $$5)\int cose{c}^{2}xdx=-\cot x+C$$ $$6)\int \sec x\tan xdx=\sec x+C$$ $$7)\int cosecxcotxdx=-cosecx+C$$ $$8)\int \frac{dx}{\sqrt{1-x^2}}={\sin }^{-1}x+C$$ $$9)\int \frac{dx}{\sqrt{1-x^2}}=-{\cos }^{-1}x+C$$ $$10)\int \frac{dx}{1+x^2}=ta{n}^{-1}x+C$$ $$11)\int (3x+5{)}^{10}dx=\frac{(3x+5{)}^{11}}{(11)(3)}$$

Integration by parts

$$\int uvdx\ne \int udx.\int vdx$$ $$\int uvdx=u\int vdx-\int [\frac{d}{dx}(u)\int vdx]dx$$ Priority for types of different types of functions is given by ILATE:
I $⟶$ Inverse Trigonometric function
L $⟶$ Logarithmic function
A $⟶$ Algebraic function
T $⟶$ Trigonometric function
E $⟶$ Exponential function

Example : $$\int x{e}^{x}dx$$ In this problem, our first function will be $x$ and second function will be $e^x$ $$=x\int e^{x}dx-\int [\frac{d}{dx}(x)\int e^{x}dx]dx$$ $$=x{e}^x-\int e^{x}dx$$ $$=x{e}^x-e^x+C$$