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Question: **.** \(\int_{0}^{\pi}\frac{x^{3}\cos^{4}x\sin^{2}x}{\pi^{2} - 3\pi x + 3x^{2}}\)dx is equal to –...

. ∫0πx3cos⁡4xsin⁡2xπ2−3πx+3x2\int_{0}^{\pi}\frac{x^{3}\cos^{4}x\sin^{2}x}{\pi^{2} - 3\pi x + 3x^{2}}dx is equal to –

A

π216\frac{\pi^{2}}{16}

B

π28\frac{\pi^{2}}{8}

C

π24\frac{\pi^{2}}{4}

D

π232\frac{\pi^{2}}{32}

Answer

π232\frac{\pi^{2}}{32}

Explanation

Solution

Let I = dx

= ∫0π(π−x)3cos⁡4(π−x)sin⁡2(π−x)π2−3π(π−x)+3(π−x)2\int _ { 0 } ^ { \pi } \frac { ( \pi - x ) ^ { 3 } \cos ^ { 4 } ( \pi - x ) \sin ^ { 2 } ( \pi - x ) } { \pi ^ { 2 } - 3 \pi ( \pi - x ) + 3 ( \pi - x ) ^ { 2 } }dx

= dx

= dx – I

Ž 2I = pdx

\ I = p . sin2 x dx = 2p .

\ =π232\frac { \pi ^ { 2 } } { 32 }.

Hence (4) is the correct answer.