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Can you do the 6 steps and show the work please

To solve these integrals using **u-substitution**, we will go through the standard 6 steps:

1. **Choose the substitution** \( u = f(x) \)
2. **Find the derivative of \( u \)** (i.e., \( du \))
3. **Rewrite the integral in terms of \( u \)**
4. **Perform the integration in terms of \( u \)**
5. **Substitute back \( u = f(x) \)**
6. **Simplify the result**

### Let's solve the integrals step-by-step:

### (a) \( \int \left( \sqrt{x^2 + 2x} \cdot (2x + 2) \right) \, dx \)

#### Step 1: Choose the substitution.
- We will choose the expression inside the square root as \( u \). 
  Let \( u = x^2 + 2x \).

#### Step 2: Find the derivative of \( u \) (i.e., \( du \)).
- Differentiate \( u = x^2 + 2x \) with respect to \( x \):
  \[
  du = (2x + 2) \, dx
  \]
  So, \( 2x + 2 \, dx = du \), which is exactly what we have in the integrand.

#### Step 3: Rewrite the integral in terms of \( u \).
- Substitute \( u = x^2 + 2x \) and \( du = (2x + 2) \, dx \):
  \[
  \int \left( \sqrt{x^2 + 2x} \cdot (2x + 2) \right) dx = \int \sqrt{u} \, du
  \]

#### Step 4: Perform the integration in terms of \( u \).
- The integral of \( \sqrt{u} = u^{1/2} \) is:
  \[
  \int u^{1/2} \, du = \frac{2}{3} u^{3/2} + C
  \]

#### Step 5: Substitute back \( u = x^2 + 2x \).
- Now replace \( u \) with \( x^2 + 2x \):
  \[
  \frac{2}{3} (x^2 + 2x)^{3/2} + C
  \]

#### Step 6: Simplify the result.
- The final answer for part (a) is:
  \[
  \frac{2}{3} (x^2 + 2x)^{3/2} + C
  \]

---

### (b) \( \int \left( \frac{\cos x}{e^{\sin x}} \right) dx \)

#### Step 1: Choose the substitution.
- Let \( u = \sin x \), since \( \cos x \) is the derivative of \( \sin x \).

#### Step 2: Find the derivative of \( u \) (i.e., \( du \)).
- Differentiate \( u = \sin x \):
  \[
  du = \cos x \, dx
  \]
  So, \( \cos x \, dx = du \).

#### Step 3: Rewrite the integral in terms of \( u \).
- Substitute \( u = \sin x \) and \( du = \cos x \, dx \):
  \[
  \int \frac{\cos x}{e^{\sin x}} \, dx = \int \frac{1}{e^u} \, du
  \]

#### Step 4: Perform the integration in terms of \( u \).
- The integral of \( \frac{1}{e^u} = e^{-u} \) is:
  \[
  \int e^{-u} \, du = -e^{-u} + C
  \]

#### Step 5: Substitute back \( u = \sin x \).
- Now replace \( u \) with \( \sin x \):
  \[
  -e^{-\sin x} + C
  \]

#### Step 6: Simplify the result.
- The final answer for part (b) is:
  \[
  -e^{-\sin x} + C
  \]

---

Would you like more details on any step or clarification on the concepts? Here are 5 related questions to deepen your understanding:

1. How do you choose the right substitution in u-substitution problems?
2. What is the relationship between integration by parts and u-substitution?
3. Can all integrals be solved using u-substitution, or are there exceptions?
4. How does u-substitution simplify more complex integrals?
5. How do constants of integration factor into definite integrals with u-substitution?

**Tip:** When choosing a substitution, look for functions within the integrand whose derivative is also present (or nearly present) elsewhere in the integrand.

Can you do the 6 steps and show the work please?

This page explains how to solve integrals using u-substitution with a detailed step-by-step process, focusing on problems involving algebraic expressions and trigonometric functions. Perfect for students in Grades 11-12.

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