# Integration by Substitution (Part 2)

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

## Example 1

Just like solving indefinite integration by substitution,

change the variable.

Set sin *x* = *t*.

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

Derivative of sin *x*

Now, here's the difference between

the definite integration and the indefinite integration.

In definite integration,

you should also change the limits,

because the variable is changed.

Use *t* = sin *x* to change the limits.

If *x* = 0, then *t* = 0.

If *x* = *π*/2, then *t* = 1.

sin *x* = *t*

cos *x* *dx* = *dt*

[0, *π*/2] → [0, 1]

So (given) = ∫_{0}^{1} *e*^{t} *dt*.

Solve the integration.

Indefinite integration of *e*^{x}

## Example 2

Set ln *x* = *t*.

Then 1/*x* *dx* = *dt*.

Derivative of ln *x*

Next, use *t* = ln *x* to change the limits.

If *x* = *e*, then *t* = 1.

If *x* = *e*^{4}, then *t* = 4.

ln *x* = *t*

1/*x* *dx* *dx* = *dt*

[*e*, *e*^{4}] → [1, 4]

So (given) = ∫_{1}^{4} 1/*t* *dt*.

Solve the integration.

Indefinite integration of 1/*x*