# Integration by Substitution

Tutorials with examples and detailed solutions and exercises with answers on how to use the powerful technique of integration by substitution to find integrals.

## Review Integration by Substitution

The method of integration by substitution may be used to easily compute complex integrals. Let us examine an integral of the form Let us make the substitution u = g(x), hence du/dx = g'(x) and du = g'(x) dxWith the above substitution, the given integral is given by

In what follows C is a constant of integration which is added in the final result.

## Examples

### Example 1

Evaluate the integral
__Solution to Example 1:__

Let u = a x + b which gives du/dx = a or dx = (1/a) du. The substitution helps in computing the integral as follows

∫sin(a x + b) dx

= (1/a) ∫sin(u) du

= (1/a) (-cos(u)) + C

= - (1/a) cos(a x + b) + C

### Example 2

Evaluate the integral^{3x - 2}dx

__Solution to Example 2:__

Let u = 3x - 2 which gives du/dx = 3 or dx = (1/3) du. Hence

∫e^{3x - 2} dx

= ∫e^{u} (1/3) du

= (1/3) e^{u}

= (1/3) e^{3x - 2} + C

### Example 3

Evaluate the integral^{2}+ 5)

^{4}dx

__Solution to Example 3:__

Let u = 2x^{2} + 5 which gives du/dx = 4x, du = 4x dx, (1/4) du = x dx . The substitution gives

∫x (2x^{2} + 5)^{4} dx

= ∫(1/4) (u)^{4} du

= (1/4) (1/5) u^{5}

= (1/20) (2x^{2} + 5)^{5} + C

### Example 4

Evaluate the integral
__Solution to Example 4:__

Let u = 2x + 1 which gives du/dx = 2 and dx = (1/2) du. Solve u = 2x + 1 for x to obtain x = (1/2)(u - 1). The substitution gives

∫x √(2x + 1) dx

= ∫(1/2)(u - 1)(u)^{1/2} (1/2) du

= (1/4) ∫(u^{3/2} - u^{1/2}) du

= (1/4)( (2/5)u^{5/2} - (2/3)u^{3/2} )

= ( (2x + 1)^{3/2}(3x - 1) ) / 15 + C

### Example 5

Evaluate the integral^{-4}dx

__Solution to Example 5:__

Let u = x - 5 which gives du/dx = 1. Substituting into the given integral, we obtain

∫(x - 5)^{-4} dx

= ∫u^{-4} du

= (-1/3)u^{-3}

= (-1/3)(x - 5)^{-3} + C

### Example 6

Evaluate the integral^{x2 + 2}dx

__Solution to Example 6:__

Let u = x^{2} + 2 which gives du/dx = 2x and (1/2) du = x dx. A substitution into the given the integral gives

∫-x e^{x2 + 2} dx

= ∫- e^{u} (1/2) du

= - (1/2) ∫e^{u} du

= - (1/2) e^{u}

= - (1/2) e^{x2 + 2} + C

### Example 7

Evaluate the integral^{4}(x) dx

__Solution to Example 7:__

Let u = sin(x) which gives du/dx = cos(x) or cos(x) dx = du. Substitute into the integral to obtain

∫cos(x) sin^{4}(x) dx
= ∫u^{4} du

= (1/5) u^{5}

= (1/5) sin^{5}(x) + C

### Example 8

Evaluate the integral
__Solution to Example 8:__

Let u = 4x + 1 which gives du/dx = 4 or dx = (1/4) du. Solve u = 4x + 1 for x to obtain x = (1/4)(u - 1). Substitute to obtain

∫(3x / (4x + 1)) dx

= ∫3 (1/4) ( (u - 1) / u ) (1/4) du

= (3/16)∫(u - 1) / u du

= (3/16)∫(1 - 1/u) du

= (3/16)(u - ln|u|)

= (3/16)(4x + 1 - ln|4x + 1|) + C

### Example 9

Evaluate the integral
__Solution to Example 9:__

Let u = x - 2 which gives du/dx = 1, dx = du and x = u + 2. Substitution

∫(x / √(x - 2)) dx

= ∫(u+2) / √u) dx

= ∫(u^{1/2} + 2u^{-1/2}) dx

= (2/3)u^{3/2} + 2 * 2 u^{1/2}

= (2/3)(x - 2)^{3/2} + 4(x - 2)^{1/2} + C

### Example 10

Evaluate the integral^{3}(x + 4)

^{2}dx

__Solution to Example 10:__

Let u = x + 2 which gives du/dx = 1, dx = du and also x = u - 2. Using the above substitution we obtain

∫(x + 2)^{3}(x + 4)^{2} dx

= ∫u^{3}(u + 2)^{2} du

= ∫(u^{5} + 4u^{4} + 4u^{3})du

= (1/6) u^{6} + (4/5)u^{5} + u^{4}

= (1/6)(x + 2)^{6} + (4/5)(x + 2)^{5} + (x + 2)^{4} + C

### Example 11

Evaluate the integral^{2}+ 3x + 1) ) dx

__Solution to Example 11:__

Let u = x^{2} + 3x + 1 which gives du/dx = 2x + 3 or (2x + 3) dx = du. The substitution helps in computing the integral as follows

∫( 1 / u ) du

= ln|u|

= ln |x^{2} + 3x + 1| + C

## Exercises

Use the table of integrals and the method of integration by parts to find the integrals below. [Note that you may need to use the method of integration by parts more than once].1. ∫ cos(3x - 2) dx

2. ∫ e

^{4x - 7}dx

3. ∫ x(4x

^{2}+ 5)

^{4}dx

4. ∫ 1 / (x + 3)

^{3}dx

### Answers to Above Exercises

1. (1/3) sin(3x - 2)

2. (1/4) e

^{4x - 7}+ C

3. (1/40) (4x

^{2}+ 5)

^{5}+ C

4. (-1/2) 1 / (x + 3)

^{2}+ C