Antiderivative/Integral Help

[HWilliams44]

I want to evaluate the antiderivative of (1+4(sinx)^2)/(sinx)^2. I think I should use u substitution, and I tried making u=sinx and dx=du/cosx. I worked the problem out, and got the antiderivative of (1+4u^2)/(ucosx). Is my line of thinking correct? If so, what should I do next? Thanks.

 

[pka]

I want to evaluate the antiderivative of (1+4(sinx)^2)/(sinx)^2. I think I should use u substitution, and I tried making u=sinx and dx=du/cosx. I worked the problem out, and got the antiderivative of (1+4u^2)/(ucosx). Is my line of thinking correct? If so, what should I do next? Thanks.

There is no need for a u-substitution. Forget it.
\(\displaystyle \dfrac{{1 + 4{{\sin }^2}(x)}}{{{{\sin }^2}(x)}} = {\csc ^2}(x) + 4\)

You surely know the anti derivative of \(\displaystyle {\csc ^2}(x) + 4~???\)

 

[HWilliams44]

Oh, now I see! So, the answer would be -cotx + 4x + C. Thanks! ^.^