Evaluate the following integral: `int_0^(pi//2)1/(5cos x+3sinx)dx`

Explanation

Given,I=`int_0^(pi//2)1/(5cos x+3sinx)dx`

I=`int_0^(pi//2)1/(5(1-tan^2 x/2)/(1+tan^2x/2)+3(2tanx/2)/((1+tan^2x/2))dx`

=`int _0^(π/2)(sec^2x/2)/(​​5−5tan^2(x/2)​+6tan(x/2)​)​dx`

Let `t=tanx/2​⇒dt=1/2​sec^2 (x/2)​dx`

When x=0⇒t=0

When `x=π/2​⇒t=tan(π/4)​=1​`

=`2int _0^1 dt/(​5−5t^2+6t)`

​=`-2int _0^1 dt/(​5t^2−6t-5)`

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