Question

Choose the correct answers`int(cos2x)/((sinx+cosx)^2)dx`is equal(A) `(-1)/(sinx+cosx)+C` (B) `log|sinx+cosx|+C`C) `log|sinx-cosx|+C` (D) `1/((sinx+cosx)^2)`

Answer 1

`I = int (cos2x)/((sinx+cosx)^2)dx`
`=> I = int (cos^2x-sin^2x)/((sinx+cosx)^2)dx`
`=> I = int ((cosx+sinx)(cosx-sinx))/((sinx+cosx)^2)dx`
`=> I = int (cosx-sinx)/(sinx+cosx)dx`
Let `sinx+cosx = t` Then, ` (cosx-sinx) dx = dt`
Then,
`I = int dt/t`
`I = log |t| +c`
`I = log|sinx+cosx|+c`
So, option `B` is the correct option.