If 8x+1 2x- 2x dx=a 8x+C, then a = - Vidyadip

MCQ

If $\int\frac{\cos8\text{x}+1}{\tan2\text{x}-\cot2\text{x}}\text{ dx}=\text{a}\cos8\text{x}+\text{C},$ then $a =$

A
$-\frac{1}{16}$
B
$\frac{1}{8}$
✓
$\frac{1}{16}$
D
$-\frac{1}{8}$

✓

Answer

Correct option: C.

$\frac{1}{16}$

$\int\frac{\cos8\text{x}+1}{\tan2\text{x}-\cot2\text{x}}\text{ dx}$
$=\int\frac{2\cos^24\text{x}}{\frac{\sin2\text{x}}{\cos2\text{x}}-\frac{\cos2\text{x}}{\sin2\text{x}}}\text{ dx}$
$=\int\frac{2\cos^24\text{x}}{\sin^22\text{x}-\cos^22\text{x}}\times\sin2\text{x}\cos2\text{x dx}$
$=\int-\frac{\cos^24\text{x}\sin4\text{x}}{\cos4\text{x}}\text{ dx}$
$=\frac{-1}{2}\int\sin8\text{x dx}$
$=\frac{\cos8\text{x}}{16}+\text{C}$
$\text{a}=\frac{1}{16}$

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