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

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 ：\(\boxed{I = \frac{1}{12} (2 + 2x^2)^3 + C}\)