∫4sin⁡x⋅cos⁡x2⋅cos⁡3x2dx is equal to

I=∫4sin⁡x⋅cos⁡x2⋅cos⁡3x2dx=∫2sin⁡x(cos⁡2x+cos⁡x)dx=∫(sin⁡3x−sin⁡x+sin⁡2x)dx=cos⁡x−13cos⁡3x−12cos⁡2x+C

∫4sin⁡x⋅cos⁡x2⋅cos⁡3x2dx is equal to

I=∫4sin⁡x⋅cos⁡x2⋅cos⁡3x2dx=∫2sin⁡x(cos⁡2x+cos⁡x)dx=∫(sin⁡3x−sin⁡x+sin⁡2x)dx=cos⁡x−13cos⁡3x−12cos⁡2x+C

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$\int 4 \sin x \cdot \cos \frac{x}{2} \cdot \cos \frac{3 x}{2} d x is equal to$

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$\cos x + \frac{1}{2} \cos 2 x + \frac{1}{3} \cos 3 x + C$

$\cos x + \frac{1}{2} \cos 2 x - \frac{1}{3} \cos 3 x + C$

$\cos x - \frac{1}{2} \cos 2 x + \frac{1}{3} \cos 3 x + C$

$\cos x - \frac{1}{2} \cos 2 x - \frac{1}{3} \cos 3 x + C$

answer is B.

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Detailed Solution

$\begin{aligned} I = \int 4 \sin x \cdot \cos \frac{x}{2} \cdot \cos \frac{3 x}{2} dx \\ = \int 2 \sin x (\cos 2 x + \cos x) dx \\ = \int (\sin 3 x - \sin x + \sin 2 x) dx \\ = \cos x - \frac{1}{3} \cos 3 x - \frac{1}{2} \cos 2 x + C \end{aligned}$

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∫4sin⁡x⋅cos⁡x2⋅cos⁡3x2dx is equal to

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$\int 4 \sin x \cdot \cos \frac{x}{2} \cdot \cos \frac{3 x}{2} d x is equal to$