Can you integrate?

Calculus Level 3

F i n d ∫ 0 π sin ⁡ x ( cos ⁡ x ) 2 d x Find\quad \int _{ 0 }^{ \pi }{ \sin { x } { (\cos { x } ) }^{ 2 } } dx


The answer is 0.66666.

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Rajdeep Dhingra by Rajdeep Dhingra , 20, India

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3 solutions

let cosx=t ,-sinxdx=dt and limits change to cos0=1 and cos pi=-1. integration of [-t^2 dt] with limits 1 to -1. on solving we get -t^3/3. after putting limits 2/3=0.667

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Yeah so easy!

Kartik Sharma - 6 years, 7 months ago

Thanks for the solution

Rajdeep Dhingra - 6 years, 8 months ago

∫ 0 π s i n x ( c o s x ) 2 = − 1 3 ( c o s 3 x ) 0 π = 2 3 \quad\int_{0}^{\pi}sinx(cosx)^{2} =- \frac{1}{3}(cos^{3}x)_{0}^{\pi}=\frac{2}{3}

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James Wayle
Jul 13, 2015

Consider y=Cos^3(x), dy/dx by the chain rule is -3Sin(x)Cos^2(x), adjusting for the intended result in this question we now consider y= -(1/3)Cos^3(x), which is the desired solution as constant coefficients are invariant. Next substitute for the limits in radians and simplify.

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