Integral: Solve $\int \frac{\sin x+\cos x}{\sec x+ \tan x}dx$

[alba_ei]

Homework Statement

\int \frac{\sin x+\cos x}{\sec x+ \tan x}dx

Homework Equations

\sin x = \frac{1}{\sec x}
\cos x = \frac{\sin x}{\tan x}

The Attempt at a Solution

i separate and try to use identities but i got nothign

1/(secx^2+secx tan x)+sin x/tanx^2+sec x

 

[mjsd]

write everything in terms of sin x and cos x only, then integrate with change of variable.

 

[cristo]

One of your identities is incorrect: sec(x)=1/cos(x).

As has been said above, you should first look to express everything in terms of sines and cosines. See if this gives you a hint as to how to proceed.

 

[Buddy J.]

please tell more

this is the shape that i got
don't know how to complete

–integral sin²x-sin2x-1/2(1+sinx)

 

[Gib Z]

-(\int \sin^2 x dx - \int \sin 2x dx - 1/2\int 1+\sin x)

Write sin^2 x as (1/2) (1-cos2x).

for sin 2x, make a substitution u=2x, then remember the integral of sin u is -cos u.

For the 3rd one, if you can't do it, why are you doing this question?

 

[Qyzren]

you can also try weierstrass substitution
i.e t = tan x/2