Using the method of substitution,
\[
I=\int x^{1}\left(2+2 x^{2}\right)^{2} d x=\int f(u) d u
\]
where
\[
\begin{array}{l}
u=\square \\
d u=\square d x \\
f(u)=\square
\end{array}
\]
Using the above information:
\[
\begin{array}{l}
\int x^{1}\left(2+2 x^{2}\right)^{2} d x \\
=\int f(u) d u \\
=\square \text { (in terms of } u \text { ) } \\
=\square \text { (in terms of } x \text { ) }
\end{array}
\]
Note: Don't forget the constant of integration in the last two blanks
Answer
\(\boxed{I = \frac{1}{12} (2 + 2x^2)^3 + C}\)
Step 1 ：Let \(u = 2 + 2x^2\)
Step 2 ：Differentiate \(u\) with respect to \(x\) to find \(du\), \(du = 4x \, dx\)
Step 3 ：Rearrange to find \(dx\), \(dx = \frac{du}{4x}\)
Step 4 ：Substitute \(u\) and \(dx\) into the integral, \(I = \int x \cdot u^2 \cdot \frac{du}{4x}\)
Step 5 ：Cancel out the \(x\) terms, \(I = \frac{1}{4} \int u^2 \, du\)
Step 6 ：Solve the integral using the power rule, \(I = \frac{1}{4} \cdot \frac{u^3}{3} + C\)
Step 7 ：Substitute \(u = 2 + 2x^2\) back in, \(I = \frac{1}{12} (2 + 2x^2)^3 + C\)
Step 8 ：\(\boxed{I = \frac{1}{12} (2 + 2x^2)^3 + C}\)
