# The integral of cosine. How to integrate cos(ax+b), eg cos2x = 1/2sin2x + c

The integral of cosine is sine:

Therefore:

∫cosdx = sinx + c

However, this is probably a more useful result to remember when integrating:

∫cos(ax + b) = (1/a)sin(ax+b) + c

Let’s take a look at a few examples of working out the integral of cosine.

**Example 1**

Work out ∫cos(5x+3)dx

So cosine integrated goes to sine and multiply the front by 1/5 (since the coefficient of x is 5 and 1 ÷ 5 is ⅕):

∫cos(5x+3)dx = ⅕sin(5x+3) + c

**Example 2**

Work out ∫cos(8x)dx

Once again, cosine integrated goes to sine and multiply the front by 1/8 (since the coefficient of x is 8 and 1 ÷ 8 = ⅛):

∫cos(8x)dx = ⅛sin(8x) + c

**Example 3**

Work out ∫3cos(2x + 4)dx

First of all take the coefficient of 3cos(2x+4) outside the integral before you integrate:

∫3cos(2x + 4)dx = 3∫cos(2x+4)

Once again, cosine integrated goes to sine and multiply the front by 1/2 (since the coefficient of x is 2 and 1 ÷ 2 = ½):

= 3 × ½sin(2x+4) + c

= 3/2sin(2x+4) + c

**Example 4**

Work out ∫5cos(⅓x - 20)dx

First of all take the coefficient of 5cos(⅓x - 20) outside the integral before you integrate:

∫5cos(⅓x - 20)dx = 5∫cos(⅓x - 20)

Once again, cosine integrated goes to sine and multiply the front by 3 (since 1 ÷ 1/3 = 3):

= 5 × 3sin(⅓x - 20) + c

= 15sin(⅓x - 20) + c

**Example 5**

Work out ∫-cos(6x)dx

First of all take the coefficient of -cos(⅓x - 20) outside the integral before you integrate:

∫-cos(6x)dx = -1∫cos(6x)

Once again, cosine integrated goes to sine and multiply the front by 1/6 (since 1 ÷ 6 = 1/6):

= -1 × ⅙sin(6x) + c

= -⅙sin(6x) + c

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