It appears that the image did not yield any text through my initial attempt to extract it using OCR. I'll instead manually solve the integral problems based on the visual data.

Problem 19:

We need to evaluate the integral: $\int_1^2 \left( e^{4u} - \frac{1}{(u+1)^2} \right) du$

Step 1: Break the integral into two parts: $\int_1^2 e^{4u} du - \int_1^2 \frac{1}{(u+1)^2} du$

Step 2: Solve each part.

1. For the first part: $\int e^{4u} du = \frac{e^{4u}}{4}$ Now evaluate from 1 to 2: $\left[\frac{e^{4u}}{4}\right]_1^2 = \frac{e^8}{4} - \frac{e^4}{4}$

2. For the second part: $\int \frac{1}{(u+1)^2} du = -\frac{1}{u+1}$ Now evaluate from 1 to 2: $\left[ -\frac{1}{u+1} \right]_1^2 = -\frac{1}{3} + \frac{1}{2} = \frac{1}{6}$

Final result: Combining both parts: $\frac{e^8}{4} - \frac{e^4}{4} + \frac{1}{6}$

Problem 21:

We need to evaluate the integral: $\int_{-1}^0 y(2y^2 - 3)^5 dy$

Let $u = 2y^2 - 3$, which simplifies the integral.

Step 1: Differentiate $u$: $du = 4y dy \quad \Rightarrow \quad y dy = \frac{1}{4} du$

Step 2: Change the limits of integration.

• When $y = -1$, $u = 2(-1)^2 - 3 = -1$.
• When $y = 0$, $u = -3$.

Thus, the integral becomes: $\frac{1}{4} \int_{-1}^{-3} u^5 du$

Step 3: Solve the new integral: $\frac{1}{4} \int u^5 du = \frac{1}{4} \left( \frac{u^6}{6} \right) = \frac{u^6}{24}$

Step 4: Evaluate the integral from -1 to -3: $\left[ \frac{u^6}{24} \right]_{-1}^{-3} = \frac{(-3)^6}{24} - \frac{(-1)^6}{24} = \frac{729}{24} - \frac{1}{24} = \frac{728}{24} = \frac{182}{6}$

Thus, the value of the integral is: $\frac{182}{6}$

Would you like more details on any step or further questions? Here are 5 related questions you might explore:

1. How does substitution work for solving integrals?
2. Can you explain the steps for solving exponential integrals?
3. How would you apply integration by parts to a similar integral?
4. What are the conditions under which a definite integral converges?
5. How can you use integration to solve physics problems like finding areas or volumes?

Tip: When solving integrals, always check if substitution or breaking up the integral into simpler parts can make the process easier.