https://file.aiquickdraw.com/mathbot/file/1728097298026rifjz0eq.jpg

Please do questions 11 and 13 and show work/steps

Let's solve questions 11 and 13 step by step.

### Question 11: 
\[
\int 3x^2 e^{2x^3} \, dx
\]

#### Step 1: Substitution
Notice that the exponent \( 2x^3 \) suggests using the substitution method. Let's set:
\[
u = 2x^3 \quad \Rightarrow \quad du = 6x^2 \, dx \quad \Rightarrow \quad \frac{du}{6} = x^2 \, dx
\]

Now, rewrite the integral:
\[
\int 3x^2 e^{2x^3} \, dx = \int 3 \cdot \frac{du}{6} e^u = \frac{1}{2} \int e^u \, du
\]

#### Step 2: Integrating
The integral of \( e^u \) is straightforward:
\[
\frac{1}{2} \int e^u \, du = \frac{1}{2} e^u + C
\]

#### Step 3: Back-Substitute
Now, substitute \( u = 2x^3 \) back into the solution:
\[
\frac{1}{2} e^u + C = \frac{1}{2} e^{2x^3} + C
\]

Thus, the solution to the integral is:
\[
\boxed{\frac{1}{2} e^{2x^3} + C}
\]

---

### Question 13:
\[
\int (1 - t) e^{2t - t^2} \, dt
\]

#### Step 1: Substitution
Here, the exponent \( 2t - t^2 \) suggests using substitution. Set:
\[
u = 2t - t^2 \quad \Rightarrow \quad du = (2 - 2t) \, dt = -2(t - 1) \, dt \quad \Rightarrow \quad \frac{-du}{2} = (t - 1) \, dt
\]

Now, rewrite the integral:
\[
\int (1 - t) e^{2t - t^2} \, dt = -\frac{1}{2} \int e^u \, du
\]

#### Step 2: Integrating
The integral of \( e^u \) is again straightforward:
\[
-\frac{1}{2} \int e^u \, du = -\frac{1}{2} e^u + C
\]

#### Step 3: Back-Substitute
Substitute \( u = 2t - t^2 \) back into the solution:
\[
-\frac{1}{2} e^u + C = -\frac{1}{2} e^{2t - t^2} + C
\]

Thus, the solution to the integral is:
\[
\boxed{-\frac{1}{2} e^{2t - t^2} + C}
\]

---

### Summary of Solutions:
11. \( \boxed{\frac{1}{2} e^{2x^3} + C} \)  
13. \( \boxed{-\frac{1}{2} e^{2t - t^2} + C} \)

Would you like further explanations or details on any steps?

### Follow-up Questions:
1. How do you identify when substitution is a good method to use?
2. Why is the derivative of \( x^3 \) important in question 11?
3. How would these integrals change if the exponents were more complex, such as higher-order polynomials?
4. Can you generalize the steps for solving integrals involving \( e^{f(x)} \)?
5. What if the constant in question 11 was different, like \( e^{3x^3} \)? How would that change the approach?

#### Tip: 
When dealing with exponentials like \( e^{f(x)} \), look for a substitution that simplifies \( f(x) \) to a linear form for easier integration!

Please do questions 11 and 13 and show work/steps.

Step-by-step solution to solve integrals like ∫ 3x^2 e^(2x^3) dx and ∫ (1 - t) e^(2t - t^2) dt using substitution and exponential function integration methods. Ideal for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation