1. $[-13$ Points $]$
DETAILS SCALC9 4.5.001.EP.
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PRACTICE ANOTHER
Consider the following integral.
\[
\int \cos (4 x) d x
\]
Given the substitution $u=4 x$, find $d u$.
\[
d u=(\square) d x
\]
Rewrite the given integral in terms of $u$.
Evaluate the integral by making the given substitution. (Use $C$ for the constant of integration.)
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Answer
We evaluate the integral. The integral of $\cos(u)$ is $\sin(u)$, so the integral of $\cos(u) \frac{du}{4}$ is $\frac{\sin(u)}{4} + C$, where $C$ is the constant of integration. After substituting $u$ back in terms of $x$, the final answer is $\boxed{\frac{\sin(4x)}{4} + C}$.
Step 1 ：Given the substitution $u=4x$, we find $du$ by taking the derivative of $u$ with respect to $x$, multiplied by $dx$. Therefore, $du = 4dx$.
Step 2 ：We rewrite the given integral in terms of $u$. The integral is currently $\int \cos(4x) dx$. We substitute $u$ for $4x$, and $\frac{du}{4}$ for $dx$, to get $\int \cos(u) \frac{du}{4}$.
Step 3 ：We evaluate the integral. The integral of $\cos(u)$ is $\sin(u)$, so the integral of $\cos(u) \frac{du}{4}$ is $\frac{\sin(u)}{4} + C$, where $C$ is the constant of integration. After substituting $u$ back in terms of $x$, the final answer is $\boxed{\frac{\sin(4x)}{4} + C}$.
