Question 3 (10 points) Evaluate $$ \int \sin x \cos ^{4} x d x $$ Paragraph $$ \vee $$ B $$ \underline{U} \cup A_{d} \equiv $$

Question

UpStudy AI · Accepted Answer

We begin with the integral

$$
\int \sin x \cos^4 x \, dx.
$$

**Step 1: Choose a substitution.**

Let

$$
u = \cos x.
$$

Then, differentiate with respect to $$ x $$:

$$
\frac{du}{dx} = -\sin x \quad \Rightarrow \quad du = -\sin x \, dx.
$$

This implies

$$
\sin x \, dx = -du.
$$

**Step 2: Substitute into the integral.**

Replacing in the original integral, we have:

$$
\int \sin x \cos^4 x \, dx = \int u^4 (-du) = -\int u^4 \, du.
$$

**Step 3: Integrate with respect to $$ u $$.**

Integrate using the power rule:

$$
-\int u^4 \, du = -\frac{u^5}{5} + C.
$$

**Step 4: Substitute back in for $$ u $$.**

Replacing $$ u $$ with $$ \cos x $$, the final answer is:

$$
-\frac{\cos^5 x}{5} + C.
$$

Thus, the evaluated integral is

$$
-\frac{\cos^5 x}{5} + C.
$$
The integral evaluates to $$ -\frac{\cos^5 x}{5} + C $$.

Question 3 (10 points) Evaluate \( \int \sin x \cos ^{4} x d x \) Paragraph \( \vee \) B \( \underline{U} \cup A_{d} \equiv \)