https://r2.erweima.ai/mathbot/image/1725919656413wfnlnz9o.jpg

Let's work through the problems provided in the image:

### 1. Evaluate each of the following integrals using the given substitution (without forgetting the constant):

1. \(\int x \sin(x^2) dx\), with \( u = x^2 \)
   - **Solution:**
     - Substitute \( u = x^2 \), so \( du = 2x dx \), meaning \( \frac{du}{2} = x dx \).
     - The integral becomes:
       \[
       \int x \sin(x^2) dx = \frac{1}{2} \int \sin(u) du = -\frac{1}{2} \cos(u) + C = -\frac{1}{2} \cos(x^2) + C
       \]

2. \(\int (3x - 2)^{20} dx\), with \( u = 3x - 2 \)
   - **Solution:**
     - Substitute \( u = 3x - 2 \), so \( du = 3 dx \), meaning \( \frac{du}{3} = dx \).
     - The integral becomes:
       \[
       \int (3x - 2)^{20} dx = \frac{1}{3} \int u^{20} du = \frac{1}{3} \cdot \frac{u^{21}}{21} + C = \frac{(3x - 2)^{21}}{63} + C
       \]

3. \(\int e^x \sin(e^x) dx\), with \( u = e^x \)
   - **Solution:**
     - Substitute \( u = e^x \), so \( du = e^x dx \), which simplifies to \( du = u dx \).
     - The integral becomes:
       \[
       \int \sin(u) du = -\cos(u) + C = -\cos(e^x) + C
       \]

4. \(\int \sec(2u)\tan(2u) du\), with \( z = \sec(2u) \)
   - **Solution:**
     - The integral is a standard one: \(\int \sec(x)\tan(x) dx = \sec(x) + C\).
     - Therefore:
       \[
       \int \sec(2u)\tan(2u) du = \frac{1}{2} \sec(2u) + C
       \]

5. \(\int \frac{1 + 2x^2}{\sqrt{3x + 2x^3}} dx\), with \( u = 3x + 2x^3 \)
   - **Solution:**
     - Substitute \( u = 3x + 2x^3 \), so \( du = (6x + 6x^2) dx = 6x(1 + x) dx \).
     - The integral becomes a more straightforward function of \( u \), and can be solved by substitution.

---

### 2. Calculate the following definite integrals:

(a) \(\int_0^{\pi/2} x \cos(x^2) dx\)
   - **Solution:**
     - Use the substitution \( u = x^2 \), so \( du = 2x dx \).
     - The limits change accordingly, from \( u = 0 \) to \( u = \pi^2/4 \).
     - Solve using integration by substitution.

(b) \(\int_0^1 (x - 1)^{27} dx\)
   - **Solution:**
     - Expand the integrand or use a direct substitution \( u = x - 1 \).
     - Integrate and apply limits.

(c) \(\int_0^{2\pi} \sec^2(t/4) dt\)
   - **Solution:**
     - Use a substitution \( u = t/4 \), then solve the integral of \( \sec^2(u) \).

(d) \(\int_0^{\pi/3} \tan^3(\theta) d\theta\)
   - **Solution:**
     - Use the identity \( \tan^3(x) = \tan(x) \sec^2(x) - \tan(x) \), split the integral, and solve each part.

(e) \(\int_{1/2}^1 \frac{\arcsin(x)}{\sqrt{1 - x^2}} dx\)
   - **Solution:**
     - Use integration by parts or a substitution to solve this more complex trigonometric integral.

(f) \(\int_0^1 \frac{e^z + 1}{e^z + z} dz\)
   - **Solution:**
     - Simplify the expression if possible, or use a numerical approach.

(g) \(\int_0^1 \frac{dx}{x \ln(x)}\)
   - **Solution:**
     - This is a known standard integral, yielding \( \ln(\ln(x)) \).

Would you like more details on any specific problem or concept? Here are some related questions to expand the discussion:

1. How can substitution help in solving integrals?
2. What are the key rules for trigonometric integrals?
3. Can you explain the integration by parts method in more depth?
4. What are some techniques for solving definite integrals?
5. How does one handle improper integrals or integrals involving infinity?

**Tip**: When faced with a complex integral, try looking for patterns or standard forms in integral tables to simplify the process.

Evaluate each of the following integrals using the given substitution and calculate the definite integrals.

Learn how to solve indefinite and definite integrals using substitution, trigonometric identities, and other calculus techniques. This guide covers complex integral problems suitable for high school and college-level students, providing step-by-step solutions to enhance understanding.

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