https://file.aiquickdraw.com/mathbot/file/17273233491859z7cmbrl.jpg

The image contains two integrals that need to be evaluated using **u-substitution**.

### Problem (a):
\[
\int \sqrt{x^2 + 2x} \cdot (2x + 2) \, dx
\]

### Problem (b):
\[
\int \frac{\cos x}{\sin x \cdot e} \, dx
\]

Let's go through each step-by-step.

---

### Problem (a):

We start by simplifying the expression within the square root.

\[
x^2 + 2x = (x^2 + 2x + 1 - 1) = (x+1)^2 - 1
\]

Now, rewrite the integral:

\[
\int \sqrt{(x+1)^2 - 1} \cdot (2x + 2) \, dx
\]

Let \( u = x^2 + 2x \), then:

\[
du = (2x + 2) \, dx
\]

Thus, the integral simplifies to:

\[
\int \sqrt{u} \, du
\]

The integral of \( \sqrt{u} \) is:

\[
\int u^{1/2} \, du = \frac{2}{3} u^{3/2}
\]

Substitute back \( u = x^2 + 2x \):

\[
\frac{2}{3} (x^2 + 2x)^{3/2} + C
\]

---

### Problem (b):

We can factor out the constant \( \frac{1}{e} \) from the integral:

\[
\frac{1}{e} \int \frac{\cos x}{\sin x} \, dx
\]

The integral of \( \frac{\cos x}{\sin x} \) is a standard result. Let \( u = \sin x \), then:

\[
du = \cos x \, dx
\]

Thus, the integral becomes:

\[
\frac{1}{e} \int \frac{du}{u} = \frac{1}{e} \ln |u| + C
\]

Substitute \( u = \sin x \) back:

\[
\frac{1}{e} \ln |\sin x| + C
\]

---

### Final Answers:

- **(a)** \( \frac{2}{3} (x^2 + 2x)^{3/2} + C \)
- **(b)** \( \frac{1}{e} \ln |\sin x| + C \)

Would you like more details on any step?

Here are some related questions to explore:
1. What is the general rule for u-substitution in integrals?
2. How do you choose \( u \) in u-substitution?
3. How can you simplify more complex square roots in integrals?
4. How does the chain rule apply when differentiating integrals like in (a)?
5. What are common trigonometric integrals you should know?

**Tip:** Always check if the integral contains expressions that can simplify with substitutions before solving directly.

Evaluate integrals using u-substitution: (a) ∫√(x² + 2x)(2x + 2) dx, (b) ∫(cos(x)/sin(x)e) dx.

Solve integrals using u-substitution. This example includes solving the integrals ∫√(x² + 2x)(2x + 2) dx and ∫(cos(x)/sin(x)e) dx, covering u-substitution, trigonometric integrals, and simplification of square roots. Suitable for Grades 11-12 or early college students.

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