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

Ask by Cruz Carlson. 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 Cruz Carlson. in the United States

Mar 22,2025

UpStudy AI · Accepted Answer

Let $$ u = 2x - \frac{\pi}{2} $$. Then,
$$
\frac{du}{dx} = 2 \quad \Rightarrow \quad dx = \frac{du}{2}.
$$

**Change the limits:**

- When $$ x = \frac{\pi}{4} $$:
  $$
  u = 2\left(\frac{\pi}{4}\right) - \frac{\pi}{2} = \frac{\pi}{2} - \frac{\pi}{2} = 0.
  $$
- When $$ x = \frac{7\pi}{4} $$:
  $$
  u = 2\left(\frac{7\pi}{4}\right) - \frac{\pi}{2} = \frac{7\pi}{2} - \frac{\pi}{2} = 3\pi.
  $$

Now, the integral transforms to:
$$
\int_{\frac{\pi}{4}}^{\frac{7\pi}{4}} 4 \sin\left(2x - \frac{\pi}{2}\right) dx = \int_{0}^{3\pi} 4 \sin(u) \left(\frac{du}{2}\right) = 2\int_{0}^{3\pi} \sin(u)\,du.
$$

Next, find the antiderivative of $$\sin(u)$$:
$$
\int \sin(u)\,du = -\cos(u).
$$

By the Fundamental Theorem of Calculus:
$$
2\int_{0}^{3\pi} \sin(u)\,du = 2\left[-\cos(u)\right]_{0}^{3\pi} = 2\left[-\cos(3\pi) + \cos(0)\right].
$$

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

Thus:
$$
2\left[-(-1) + 1\right] = 2\left[1 + 1\right] = 2(2) = 4.
$$

The evaluated integral is:
$$
\boxed{4}.
$$
The integral evaluates to 4.

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

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

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