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Question

Evaluate: π/20(sinx+cosx)dx9+16sin2x

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Solution

I=sinx+cosx9+16sin2xdx

We have 1sin2x=sin2x+cos2x2sinxcosx

=(sinxcosx)2

sin2x=1(sinxcosx)2

Now multiplying by 16

16sin2x=1616(sinxcosx)2

Now 9+16sin2x=2516(sinxcosx)2

I=sinx+cosx2516(sinxcosx)2dx

Let 4(sinxcosx)=t

4(cosx+sinx)dx=dt

I=14dt25t2

=14dt52t2

=140log5+t5t+c

=140log5+4(sinxcosx)54(sinxcosx)+c

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