Evaluate $\int \frac{x^2}{(7x^3+2)^2} dx$ using u substitution

Question

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}

Answer 1

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.