`int (cos^5t) / sqrt(sint) dt` Find the indefinite integral

Given to solve,

`int cos^5 t/(sqrt(sint)) dt`

= `int ((cos^4 t)(cos t))/(sqrt(sint)) dt`

=`int ((cos^2 t)^2 (cos t))/(sqrt(sint)) dt`

=`int ((1- sin^2 t)^2 (cos t))/(sqrt(sint)) dt`

let `u = sin t ,du = cost dt`

so,

=`int ((1- u^2 )^2 (cos t))/(sqrt(u)) dt`

=` int ((1- u^2 )^2 )/(sqrt(u)) du`

=` int (1+u^4 - 2u^2)/u^(1/2) du`

`= int [u^(-1/2) +u^(7/2)-2u^(3/2)] du`

= `(u^(1/2))/(1/2) +(u^(9/2))/(9/2) -2(u^(5/2))/(5/2)`

=`(2)(u^(1/2)) +(2/9)(u^(9/2)) -(4/5)(u^(5/2))`

=` (2)((sin t )^(1/2)) +(2/9)((sin t)^(9/2)) -(4/5)((sin t)^(5/2))`

Comments

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