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

More by this Author


Comments

No comments yet.

    0 of 8192 characters used
    Post Comment

    No HTML is allowed in comments, but URLs will be hyperlinked. Comments are not for promoting your articles or other sites.


    Click to Rate This Article
    working