Use u-substitution, change of
limits and the FTC to evaluate the
integral. Do NOT use integral
features of your calculator.

Ask by Tyler Nguyen. in the United States

Mar 22,2025

Question

Use u-substitution, change of
limits and the FTC to evaluate the
integral. Do NOT use integral
features of your calculator.

Ask by Tyler Nguyen. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

We start with the integral

$$
\int_{\frac{\pi}{4}}^{\frac{7 \pi}{4}} 4 \sin \left(2 x-\frac{\pi}{2}\right) \,dx.
$$

**Step 1. Choose a substitution.**

Let

$$
u=2x-\frac{\pi}{2}.
$$

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

$$
\frac{du}{dx} = 2 \quad \Longrightarrow \quad dx=\frac{du}{2}.
$$

**Step 2. Change the limits of integration.**

When $$ x=\frac{\pi}{4} $$:

$$
u=2\left(\frac{\pi}{4}\right)-\frac{\pi}{2}=\frac{\pi}{2}-\frac{\pi}{2}=0.
$$

When $$ x=\frac{7\pi}{4} $$:

$$
u=2\left(\frac{7\pi}{4}\right)-\frac{\pi}{2}=\frac{7\pi}{2}-\frac{\pi}{2}=\frac{6\pi}{2}=3\pi.
$$

So, the new limits are from $$ u=0 $$ to $$ u=3\pi $$.

**Step 3. Rewrite the integral in terms of $$ u $$.**

Substitute into the integral:

$$
\int_{\frac{\pi}{4}}^{\frac{7\pi}{4}} 4 \sin \left(2 x-\frac{\pi}{2}\right) \,dx = \int_{0}^{3\pi} 4 \sin(u) \cdot \frac{du}{2} = \int_{0}^{3\pi} 2 \sin(u) \,du.
$$

**Step 4. Evaluate the integral using the FTC.**

Compute the antiderivative:

$$
\int 2\sin(u) \,du=-2\cos(u)+C.
$$

Now apply the limits:

$$
\left[-2\cos(u)\right]_{0}^{3\pi} = -2\cos(3\pi) - \left(-2\cos(0)\right).
$$

Recall that

$$
\cos(3\pi)=\cos(\pi)=-1 \quad \text{and} \quad \cos(0)=1.
$$

Thus,

$$
-2(-1) +2(1)= 2+2=4.
$$

The value of the integral is

$$
4.
$$
The value of the integral is 4.

Use u-substitution, change of
limits and the FTC to evaluate the
integral. Do NOT use integral
features of your calculator.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Calculus Questions

Use u-substitution, change of limits and the FTC to evaluate the integral. Do NOT use integral features of your calculator.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Calculus Questions

Use u-substitution, change of
limits and the FTC to evaluate the
integral. Do NOT use integral
features of your calculator.