Question

Integrate:

$\int \sin x^{2}dx$

Answer 1

Use Pythagorean Identities ${sin}^{2}x=21-2{cos}^{2}x$

$\int \frac{1}{2}-\int \frac{cos2x}{2}dx$

Use Sum Rule $\int f\left(x\right)+g\left(x\right)dx=\int f\left(x\right)dx+\int g\left(x\right)dx$

$\int \frac{1}{2}dx-\int \frac{cos2x}{2}dx$

Use this rule:$\int adx=ax+C$

$\frac{2}{x}-\int \frac{cos2x}{2}dx$

Use Constant Factor Rule $\int cf\left(x\right)dx=c\int f\left(x\right)dx$

$2x-21\int cos2xdx$

Use Integration by Substitution on $\int \cos 2xdx$

Let$u=2x,du=2dx,thendx=\frac{1}{2}du$

Using u and du above,rewrite$\int \cos 2xdx$

$\int \frac{cosu}{2}du$

Use Constant Factor Rule $\int cf\left(x\right)dx=c\int f\left(x\right)dx$

$\frac{1}{2}\int \cos udu$

The integral of\cos { u } is\quad sinu​

Substitute$u=2x$back into the original integral\quad ​$\frac{2sin2x}{2}$

Rewrite the integral with the completed substitution$\frac{x}{2}-\frac{sin2x}{4}$

Addconstant$\frac{x}{2}-\frac{sin2x}{4}+C$