Consider the following integral.
\[
\int \cos (8 x) d x
\]

Given the substitution $u=8 x$, find $d u$.
\[
d u=(8) d x
\]

Rewrite the given integral in terms of $u$.
\[
\int\left(\square_{x}\right) d u
\]

Evaluate the integral by making the given substitution. (Use $C$ for the constant of integration.)

Answer

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Answer

However, we need to substitute $u$ back in terms of $x$, so the final answer is: \[\boxed{\frac{1}{8}\sin(8x) + C}\]

Steps

Step 1 ：Given the substitution $u=8x$, we find that $du=8dx$.

Step 2 ：To rewrite the given integral in terms of $u$, we need to replace $dx$ with $du/8$. So, the integral becomes: \[\int \cos(u) \frac{du}{8}\]

Step 3 ：Now, we can evaluate the integral. The integral of $\cos(u)$ is $\sin(u)$, so: \[\int \cos(u) \frac{du}{8} = \frac{1}{8}\sin(u) + C\]

Step 4 ：However, we need to substitute $u$ back in terms of $x$, so the final answer is: \[\boxed{\frac{1}{8}\sin(8x) + C}\]