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I'm working on an integration by parts problem, and I'm trying to substitute to simplify the equation:

$$\int_\sqrt{\frac{\pi}{2}}^\sqrt{\pi} \theta^3 \cos(\theta^2) d\theta$$

Using the substitution rule for definite integrals, I substitute $\theta^2 = t$ and apply the same to the limits of integration:

$$\int_\frac{\pi}{2}^\pi t^\frac{3}{2} \cos(t) dt$$

However, Wolfram|Alpha tells me that I have done something wrong, as these two integrals are not equivalent. Where did I screw up?

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    When you do a substitution, in addition to changing the limits and all the instances of the variable, you also have to take care of the "differential". Here, $d\theta$; you don't just switch it to a $dt$.2011-02-22

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