# A definite integral with two different answers

I am trying to solve the following definite integral. It seems to have two different results if the integral is solved by two different methods. (Since this equation is a part of a larger system of equations), It’s important that the ambiguity of the results be removed i.e. only one result should be valid. $$\int_{0}^{\pi /2}\sin \theta \cos ^2 \theta\ d\theta$$ Method 1 Integration by substitution:

Method2 Integration by parts:

Now, it appears to me that both methods are valid even though the results obtained from them differ. Please advise **which one is the correct answer?**

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In your integration by substitution method, when you apply the substitution, you need to find what $du$ is in terms of $d\theta$ and substitute this into the integral.
$$u=\sin \theta$$ $$du=\cos \theta d \theta$$
The first method is not correct, you have forget to express $d\theta$ with $du$,
$u=sin\theta$, $du=cos\theta d\theta=\sqrt{1-u^2}du$
$\int sin\theta cos^2\theta d\theta=\int u(1-u^2){1\over\sqrt{1-u^2}}du$