Question

1
Evaluate :
dx.
1 + sin x + cos x​

Answer 1

Here is your answer in the attachment.

Answer 2

Answer:

ln( 1 + tan(x/2) ) + C

Step-by-step explanation:

We Can Re-Write  

1 + cosx = 2cos²(x/2)

sinx = 2sin(x/2)cos(x/2)

So, Now We are Left With

$\int\limits {\frac{1}{2cos^{2}(x/2) + 2sin(x/2)cos(x/2) } } \, dx$

Dividing Numerator and Denominator By cos²(x/2)

$\int\limits {\frac{sec^{2}(x/2)}{2( 1 + tan(x/2)) } } \, dx$

Taking 1 + tan(x/2) = t

Differentiating it with respect to x

We get

(sec²(x/2)dx)/2 = dt

Now Integration Becomes

=> $\int\limits {\frac{1}{t} } \, dt$

=> ln(t) + C

=> ln( 1 + tan(x/2) ) + C