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I heard that we can use $\tan(x)=y$ substitution in integrals where both sine and cosine is on even power like: $\int \sin^2(x)\cdot \cos^2(x)\, dx$.
How exactly can I use it? I know that we can solve it in other way, but I want to see how this substitution works.

mathse
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user128576
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  • Start with $x=\arctan(y)$ and the rule for a change of variable. –  May 31 '14 at 10:21
  • so tan(x)=y can't be used for those integrals? – user128576 May 31 '14 at 10:40
  • You can certainly make the substitution. I think Yves is suggesting a path you might follow to investigate it. If you wanted to substitute $x = \arctan(y)$ into that integral, what would $dx$ be? What would $\sin(\arctan(y)$ and $\cos(\arctan(y))$ look like? – coolpapa May 31 '14 at 10:50

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Observe that $\cos^2x = \frac{1}{1+\tan^2x} = \frac{1}{1+y^2}$, while $\sin^2x = 1 - \cos^2x = \frac{y^2}{1+y^2}$ and $dx = \frac{dy}{1+y^2}$. So, your integral transforms into $$ \int \frac{y^2\,dy}{(1+y^2)^3}.$$

ivanpenev
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