How to integrate cos^7 x*sinx?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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

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