Question

`intsinx/(1 + sinx)`dx = ______

Options

• sec x + tan x + x + c

• sec x - tan x + x + c

• sec x + tan x - x + c

• sec x - tan x - x + c

Answer 1

`intsinx/(1 + sinx)`dx = sec x - tan x + x + c

Explanation:

`intsinx/(1 + sinx)`dx

= `intsinx/(1 + sinx) . (1 - sinx)/(1 - sinx)dx`

= `int(sinx - sin^2x)/(1 - sin^2x)dx`

= `int(sinx - sin^2x)/(cos^2x)dx`

= `int(sinx/cos^2x - sin^2x/cos^2x)dx`

= `int(1/cosx . sinx/cosx - tan^2x)dx`

= `int[secx tanx - (sec^2x - 1)]dx`

= `int secx tanx dx - intsec^2x dx + int1 . dx`

= sec x - tan x + x + c