Question

Integrate (cos x - sin x)/(1 + sin 2x) dx

Answer 1

Given:

An integral

`\int(Cosx-Sinx)/(1+sin2x)dx`

Required:

Solve the integral.

Step-by-step explanation:

We will start with

`\begin{gathered} 1+Sin2x=Sin^2x+Cos^2x+2SinxCosx \\ 1+Sin^2x=(Sinx+Cosx)^2 \end{gathered}`

Now,

`=\int(Cosx-Sinx)/((Cosx+Sinx)^2)dx`

Put

`\begin{gathered} Cosx+Sinx=t \\ dt=(Cosx-Sinx)dx \end{gathered}`
`\begin{gathered} =\int(dt)/(t^2) \\ =(t^(-2+1))/(-2+1)+c \\ =(t^(-1))/(-1)+c \\ =-(1)/(t)+c \\ =-(1)/(Cosx+Sinx)+c \end{gathered}`

Answer:

Hence, above is the answer.