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**Homework Statement**

$$ \int x^{3}cos(x^{2})dx$$

**The attempt at a solution**

OK, so I am aware that there is a way in which to do this problem where you do a substitution (let $$u=x^{2}$$ to do a substitution before you integrate by parts), and I was able to get the answer right using this method. The thing is, I tried it a different way first, and after triple checking it, I feel like I should have gotten it right. It doesn't matter what I choose to let

*u*and

*dv*equal, right? I should get the right answer no matter what I choose (assuming I did things correctly, of course).

Here's my work:

$$u=cos(x^{2})$$

$$du=-2xsin(x^{2})$$

$$v=\frac{x^{4}}{4}$$

$$dv=x^{3}dx$$

$$\frac{x^{4}cos(x^{2})}{4}+\frac{1}{2}\int x*sin(x^{2})dx$$

$$u_2=sin(x^{2})$$

$$du_2=2xcos(x)$$

$$v_2=\frac{x^{2}}{2}$$

$$dv_2=x$$

$$\frac{x^{4}cos(x^{2})}{4}+\frac{1}{2}(\frac{x^{2}sin(x^{2})}{2}-\int x^{3}cos(x^{2})dx)$$

Since this last integral is the same as the one we started with, we can now say that

$$I=A+B-\frac{1}{2}I$$

Hence

$$\frac{3}{2}I=A+B$$

Multiplying both sides by two over three...

$$\frac{x^{4}cos(x^{2})}{6}+\frac{x^{2}sin(x^{2})}{6}$$

Thanks for your time and any help you can offer me.