Question

cos x17[Hint : Put sin 1-1(1-sin x) (2- sin x)

Answer 1

The integral is given as follows,

I= ∫ cosxdx ( 1−sinx )( 2−sinx )

Assume sinx=t

Differentiate the above with respect to t.

cosxdx=dt

Substitute the values in the integral.

I= ∫ cosxdx ( 1−sinx )( 2−sinx ) I= ∫ dt ( 1−t )( 2−t )

Use partial fraction rule.

1 ( 1−t )( 2−t ) = A ( 1−t ) + B ( 2−t ) 1=A( 2−t )+B( 1−t )

Substitute t=1 then,

A=1

Again substitute t=2then,

B=−1

On integrating, we get

I= ∫ dt ( 1−t )( 2−t ) = ∫ dt ( 1−t ) − ∫ dt ( 2−t ) =−log| 1−t |−( −log| 2−t | )+C =log| 1 1−t |+log| 2−t |+C

By substituting sinx for t, we get

I=log| 1 1−sinx |+log| 2−sinx |+C I=log| 2−sinx 1−sinx |+C