∫(xsinx)^2d积分限是0到π,怎么求?

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∫(xsinx)^2d积分限是0到π,怎么求?
∫(xsinx)^2d积分限是0到π,怎么求?

∫(xsinx)^2d积分限是0到π,怎么求?
∫[0,π]xsin²x dx
= ∫[0,π]x*(1/2)(1-cos2x) dx
= (1/2)∫[0,π]x dx - (1/2)∫[0,π]xcos2x dx
= (1/2)(x²/2) - (1/2)(1/2)∫[0,π]x dsin2x
= (1/4)(π²) - (1/4)xsin2x + (1/4)∫[0,π]sin2x dx
= π²/4 - (1/4)(0) - (1/4)(1/2)cos2x
= π²/4 - 0
= π²/4
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