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Question

Evaluate π20sinϕcos5ϕdϕ

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Solution

I=π20sinϕcos5ϕdϕ

=π20sinϕcos4ϕcosϕdϕ

=π20sinϕ(1sin2ϕ)2cosϕdϕ

Let t=sinϕdt=cosϕdϕ

When ϕ=0t=0 and When ϕ=π2t=1

=10t(1t2)2dt

=10t(1+t42t2)dt

=10(t12+t12+42t12+2)dt

=10(t12+t922t52)dt

=⎢ ⎢ ⎢t12+112+1+t92+192+12t52+152+1⎥ ⎥ ⎥10

=⎢ ⎢ ⎢t3232+t1121122t7272⎥ ⎥ ⎥10

=[23t32+211t11247t72]10

=[23(10)+211(10)47(10)]

=23+21147=154+42132231=64231

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