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.