How to integrate cos^7 x*sinx?

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