# Integral by Substitution: Definite Integral

How to solve the given definite integral by using the integral by substitution: 2 examples and their solutions.

## Example 1

### Example

You might want to set sin x = t,

which you've learned before.

Integral by Substitution: Indefinite Integral

The difference is the upper/lower limits.

Let's see how to solve this definite integral

by using the integral by substitution.

### Solution

Set sin x = t.

Differentiate both sides.

The derivative of sin x is cos x dx.

And the derivative of t is dt.

cos x dx is already in the given integral.

So you don't have to change cos x dx = dt.

The variable is changed from x to t.

So the upper/lower limits are also changed.

To find the changed limits,

put π/2 and 0

into t = sin x.

If x = 0,

then t = 0.

If x = π/2,

then t = 1.

sin x = t

cos x dx = dt

x = 0 → t = 0

x = π/2 → t = 1

Put these into the given integral:

∫_{0}^{π/2} e^{sin x} cos x dx.

Then (given) = ∫_{0}^{1} e^{t} dt.

Solve the integral.

Definite Integral: How to Solve

The integral of e^{t} is itself: e^{t}.

Put 1 and 0

into e^{t}.

Then e^{1} - e^{0}.

e^{1} - e^{0} = e - 1

So e - 1 is the answer.

## Example 2

### Example

### Solution

Set ln x = t.

Differentiate both sides.

The derivative of ln x is [1/x] dx.

And the derivative of t is dt.

[1/x] dx is already in the given integral.

So you don't have to change [1/x] dx = dt.

The variable is changed from x to t.

So the upper/lower limits are also changed.

To find the changed limits,

put e and e^{4}

into t = ln x.

If x = e,

then t = 1.

If x = e^{4},

then t = 4.

ln e^{4} = 4 ln e

Logarithm of a Power

ln x = t

[1/x] dx = dt

x = e → t = 1

x = e^{4} → t = 4

Put these into the given integral:

∫_{e}^{e4} dx/[x ln x].

Then (given) = ∫_{1}^{4} 1/t dt.

Solve the integral.

The integral of 1/t is

ln |t|.

Put 4 and 1

into ln |t|.

Then ln |4| - ln |1|.

ln |4| = ln 4 = ln 2^{2}

ln |1| = ln 1 = 0

ln 2^{2} = 2 ln 2

So 2 ln 2 is the answer.