Questions: Find the indefinite integral. (Use C for the constant [ int 16 x(8 x^2+2)^2 d x ]

Solution Steps

To solve the indefinite integral \(\int 16 x (8 x^{2} + 2)^{2} \, dx\), we can use the method of substitution. Let \( u = 8x^2 + 2 \), then \( du = 16x \, dx \). This substitution simplifies the integral into a basic power rule form.

We start with the integral

\[ \int 16 x (8 x^{2} + 2)^{2} \, dx. \]

We use the substitution \( u = 8x^2 + 2 \). Then, the derivative \( du = 16x \, dx \) allows us to rewrite the integral in terms of \( u \).

Substituting \( u \) into the integral, we have:

\[ \int u^{2} \, du. \]

The integral of \( u^{2} \) is

\[ \frac{u^{3}}{3} + C. \]

Substituting back \( u = 8x^2 + 2 \), we get:

\[ \frac{(8x^2 + 2)^{3}}{3} + C. \]

Expanding \( (8x^2 + 2)^{3} \) gives:

\[ (8x^2 + 2)^{3} = 512x^6 + 3 \cdot 128x^4 + 3 \cdot 32x^2 + 8. \]

Thus, the integral becomes:

\[ \frac{512x^6 + 384x^4 + 96x^2 + 8}{3} + C. \]

Final Answer