Determine the antiderivative of y=sin x*cos^n x?

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the anti derivative of y = sin x*(cos x)^n

let cos x = y

=> dy/dx = -sin x

=> (-1)*dy = sin x dx

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

=> Int [ (-1)* y^n]

=> (-1)* y^(n+1) / (n + 1)

substitute y = cos x

=> [-1/(n+1)]*(cos x)^(n+1)

The required anti derivative is [-1/(n+1)]*(cos x)^(n+1)

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To determine the antiderivative of a function, we'll have to calculate the indefinite integral of the function (cos x)^n*sin x.

Int (cos x)^n*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 n-th power cos x and the variable t:

(cos x)^n  = t^n

We'll re-write the integral:

-Int t^n dt = -t^(n+1)/(n+1) + C

We'll substitute t by cos x:

Int [(cos x)^n]*sin x dx = -(cos x)^(n+1)/(n+1) + C

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