Evaluate the integral (trig substitution)

goosefraba

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Jan 25, 2011
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I need help with the following problem please:

integral of: x^3 (sqrt(49-x^2)) dx

Any help would be greatly appreciated. Thanks
 
I suppose by your title that you have to use trig sub.

In that event, let x=7sinθ,   dx=7cosθdθ\displaystyle x=7sin{\theta}, \;\ dx=7cos{\theta}d{\theta}

Make the subs and simplify it down.

(7sinθ)349(7sinθ)27cosθdθ\displaystyle \int(7sin{\theta})^{3}\sqrt{49-(7sin{\theta})^{2}}\cdot 7cos{\theta}d{\theta}

343sin3θ49(1sin2θ)7cosθdθ\displaystyle \int 343sin^{3}{\theta}\sqrt{49(1-sin^{2}{\theta})}\cdot 7 cos{\theta}d{\theta}

16807sin3θcos2θdθ\displaystyle 16807\int sin^{3}{\theta}cos^{2}{\theta}d{\theta}

Split off a factor of sin and apply the identity sin2θ=1cos2θ\displaystyle sin^{2}{\theta}=1-cos^{2}{\theta}
 
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