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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