Question

(cos2x)/(sinx + cosx)².dx
Integrate the function

Answer 1

HELLO DEAR,

GIVEN FUNCTION IS cos2x/(sinx + cosx)².dx

so, $\sf{\int{\frac{cos2x}{(sin^2x + cos^2x + 2sinxcosx)}}\,dz}$

$\sf{\Rightarrow \int{\frac{cos2x}{1 + sin2x}}\,dx}$

let (1 + sin2x) = t , dx = dt/2cos2x

so, $\sf{\int{\frac{cos2x}{(1 + sin2x)}*dt/cos2x}\,dx}$

$\sf{\Rightarrow\int{\frac{dt}{t}}}$

$\sf{\Rightarrow log|t| + c}$

put the value of t

$\sf{\Rightarrow log|1 + sin2x| + c}$

$\sf{\Rightarrow log|(sin^2x + cos^2x + 2sinxcosx)| + c}$

$\sf{\Rightarrow log|(sinx + cosx)^2| + c}$

$\sf{\Rightarrow 2log|sinx + cosx| + c}$

I HOPE ITS HELP YOU DEAR,
THANKS