# Integration by Substitution (Part 1)

How to solve the integration by substitution problems (indefinite integration): examples and their solutions.

## Example 1

Set 2*x* - 1 = *t*.

Differentiate both sides:

2 *dx* = *dt**dx* = (1/2)*dt*.

Implicit differentiation

Substitute [2*x* - 1] with [*t*].

And substitute [*dx*] with [(1/2)*dt*].

Then (given) = ∫ *t*^{8} (1/2)*dt*.

Put 2*x* - 1 in *t*.

Then (given) = (1/18)(2*x* - 1)^{9} + *C*.

## Example 2

Set sin *x* = *t*.

Differentiate both sides:

cos *x* *dx* = *dt*.

Implicit differentiation

Derivative of sin *x*

Substitute [sin *x*] with [*t*].

And substitute [cos *x* *dx*] with [*dt*].

Then (given) = ∫ *t*^{5} *dt*.

Put sin *x* in *t*.

Then (given) = (1/6) sin^{6} *x* + *C*.