Q113 / 160 · AP EAMCET 2024 (22 May Shift 1) · Mathematics

$\int \frac{\sin 7 x}{\sin 2 x \sin 5 x} d x=$

  • A
    $\log (\sin 5 x \sin 2 x)+c$
  • B
    $\log \sin 5 x+\sin 2 x+c$
  • Correct answer
    C
    $\frac{1}{5} \log \sin 5 x+\frac{1}{2} \log \sin 2 x+c$
  • D
    $\frac{1}{5} \log \sin x+\frac{1}{2} \log \sin x+c$

Solution

$\begin{aligned} & I=\int \frac{\sin 7 x}{\sin 2 x \sin 5 x} d x \Rightarrow I=\int \frac{\sin (5 x+2 x)}{\sin 2 x \sin 5 x} d x \\ & \Rightarrow I=\int \frac{\sin 5 x \cos 2 x+\cos 5 x \sin 2 x}{\sin 2 x \sin 5 x} d x \\ & \Rightarrow I=\int \cot 2 x d x+\int \cot 5 x d x \\ & \Rightarrow I=\frac{1}{2} \log (\sin 2 x)+\frac{1}{5} \log (\sin 5 x)+c .\end{aligned}$

Hence, the correct answer is option: 'C'.

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