Calculus Questions and Answers

Start Your Free Trial

How to integrate cos^7 x*sinx?

Expert Answers info

Tushar Chandra eNotes educator | Certified Educator

calendarEducator since 2010

write12,551 answers

starTop subjects are Math, Science, and Business

We need to determine the integral of (cos x)^7 * sin x.

Int [(cos x)^7 * sin x dx]

let cos x = u => - du = sin x dx

=> Int [ -u^7 du]

=> -u^8 / 8 + C

substitute u  = cos x

=> - (cos x)^8 / 8 + C

The required integral is - (cos x)^8 / 8 + C

check Approved by eNotes Editorial

giorgiana1976 | Student

So, we'll have to calculate the indefinite integral of the function (cos x)^7*sin x.

Int (cos x)^7*sin x dx

We'll solve the indefinite integral using substitution technique.

We'll put cos x = t =>-sin x dx = dt

We'll raise to 7th power cos x:

(cos x)^7  = t^7

We'll re-write the integral:

-Int t^7 dt = -t^8/8 + c

We'll substitute t by cos x:

Int (cos x)^7*sin x dx = -(cos x)^8/8 + C