Evaluate the integral
\[
\int(2 x+4)\left(x^{2}+4 x+8\right)^{4} d x
\]
by making the substitution $u=x^{2}+4 x+8$.
Answer
Finally, we substitute $u$ back into the integral to get the final answer in terms of $x$. The integral of $(2x+4)(x^{2}+4x+8)^{4}$ with respect to $x$ is $\frac{x^9}{9} + 2x^8 + \frac{128x^7}{7} + \frac{320x^6}{3} + \frac{2176x^5}{5} + 1280x^4 + \frac{8192x^3}{3} + 4096x^2 + 4096x + C$.
Step 1 ：Let's start by making the substitution $u=x^{2}+4 x+8$. This simplifies the integral to $\int u^{4} du$.
Step 2 ：Next, we find the derivative of $u$, which is $du=(2x+4)dx$. This allows us to rewrite the integral in terms of $u$.
Step 3 ：Now, we can solve the integral $\int u^{4} du$ which is $\frac{u^5}{5} + C$.
Step 4 ：Finally, we substitute $u$ back into the integral to get the final answer in terms of $x$. The integral of $(2x+4)(x^{2}+4x+8)^{4}$ with respect to $x$ is $\frac{x^9}{9} + 2x^8 + \frac{128x^7}{7} + \frac{320x^6}{3} + \frac{2176x^5}{5} + 1280x^4 + \frac{8192x^3}{3} + 4096x^2 + 4096x + C$.
