Question

$\int \frac{3cosx}{2cosx+5sinx}dx$ is equal to
[Note : where $C$ is integration constant]

• A. $\frac{15}{29}x+\frac{6}{29}ln\left|2cosx+5sinx\right|+C$
• B. $\frac{6}{29}x-\frac{15}{29}ln\left|2cosx+5sinx\right|+C$
• C. $\frac{6}{29}x+\frac{15}{29}ln\left|2cosx+5sinx\right|+C$
• D. None of these

Answer 1

Answer: (C)

Apply $N^r=A\left(D^r\right)+B\left(\frac{d}{dx}{D}^r\right),$
$3cosx=A\left(2cosx+5sinx\right)+B\left(-2sinx+5cosx\right)$
$A=\frac{6}{29},B=\frac{15}{29}$
$\int \frac{3cosx}{2cosx+5sinx}dx=\frac{6}{29}\int dx+\frac{15}{29}\int \frac{-2sinx+5cosx}{2cosx+5sinx}dx$
$\Rightarrow I=\frac{6}{29}x+\frac{15}{29}ln\left|2cosx+5sinx\right|+C$