<s>[INST] Evaluate $\int \frac{x^2}{(7x^3+2)^2} dx$ using u substitution Section 5.2
Can somebody verify my solution? Thanks!!
Evaluate $\int \frac{x^2}{(7x^3+2)^2} dx$ using u substitution

Let $u=7x^3+2$. Then $\frac{du}{dx}=21x^2$ and so $\frac{du}{21x^2}=dx$. 
Thus we have:
\begin{align}
& \int \frac{x^2}{(7x^3+2)^2} \, dx \\[8pt]
= {} & \int \frac{x^2}{u^2} \frac{du}{21x^2}\,dx \\[8pt]
= {} & \frac{1}{21} \int \frac{1}{u^2} \, du \\[8pt]
= {} & \frac{1}{21} \int u^{-2} \, du \\[8pt]
= {} & \frac{1}{21} \frac{u^{-1}}{-1}+C \\[8pt]
= {} & \frac{-1}{21} u^{-1}+C \\[8pt]
= {} & \frac{-1}{21} (7x^3+2)^{-1}+C \\[8pt]
= {} & \frac{-1}{21(7x^3+2)}+C
\end{align}
 [/INST] I worked it out and I got the same answer you did.
One thing to point out: When you have $$\dfrac {du}{dx} = 21 x^2$$ you can write the subsitution as either $$\dfrac {du}{21 dx} = x^2$$ if you prefer to use $\dfrac {du}{dx}$ notation or $$\dfrac {1}{21}du = x^2 dx$$ if you prefer differential notation. 
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