The value of ⁡x dx is equal to.

JEE MainMathematicsModerateDefinite Integration

The value
of $\int_{0}^{2\pi}\mspace{2mu} x\sin^{6}x{\ cos}^{4}$ ⁡x dx is equal to.

Show answer and solution

I = \int_{0}^{2\pi}\mspace{2mu} x\sin^{6}x\cos^{4}xdx

I = \int_{0}^{2\pi}\mspace{2mu}(2\pi - x)\sin^{6}x\cos^{4}xdx

Add

2I = 2\pi\int_{0}^{2\pi}\mspace{2mu}\sin^{6}x\cos^{4}xdx

2I = 4\pi\int_{0}^{\pi}\mspace{2mu}\sin^{6}x\cos^{4}xdx

I = 4\pi\int_{0}^{\pi/2}\mspace{2mu}\sin^{6}x\cos^{4}xdx

= 4\pi\left( \frac{(5 \times 3 \times 1)(3 \times 1)}{10 \times 8 \times 6 \times 4 \times 2} \right) \times \frac{\pi}{2} = \frac{3\pi^{2}}{128}

Source: Grass-G / M-13 Mathematics Paper 4 Nov.docx

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