Question

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

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

Answer 1

Answer: (C)

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