Question

Solve the integral

I've come across some difficulties with this one..

Answer 1

Multiply the integrand's numerator and denominator by
`1-\sin x`, then simplify:

`\displaystyle\int(\mathrm dx)/(1+\sin x)=\int(1-\sin x)/(1-\sin^2x)\,\mathrm dx=\int(1-\sin x)/(\cos^2x)\,\mathrm dx=\int(\sec^2x-\sec x\tan x)\,\mathrm dx`

You should be familiar with

`(\mathrm d)/(\mathrm dx)\tan x=\sec^2x`

`(\mathrm d)/(\mathrm dx)\sec x=\sec x\tan x`

so that

`\displaystyle\int(\mathrm dx)/(1+\sin x)=\tan x-\sec x+C`