The image contains two indefinite integrals to solve:

A. $\int 132x \cos(9 + 11x^2) \, dx$

B. $\int 4 \cos(x)(6 + \sin(x))^7 \, dx$

Let's approach them one by one.

A. $\int 132x \cos(9 + 11x^2) \, dx$

For this, we can use u-substitution.

Let $u = 9 + 11x^2$, so that $du = 22x \, dx$.

Now, rewrite the integral:

$\int 132x \cos(9 + 11x^2) \, dx = \int \frac{132}{22} \cos(u) \, du = 6 \int \cos(u) \, du$

Now, integrate $\cos(u)$:

$6 \sin(u) + C$

Substitute back for $u$:

$6 \sin(9 + 11x^2) + C$

B. $\int 4 \cos(x)(6 + \sin(x))^7 \, dx$

Here, we use u-substitution again.

Let $u = 6 + \sin(x)$, so $du = \cos(x) \, dx$.

Now, rewrite the integral:

$\int 4 \cos(x)(6 + \sin(x))^7 \, dx = 4 \int u^7 \, du$

Now, integrate $u^7$:

$4 \cdot \frac{u^8}{8} + C = \frac{1}{2} u^8 + C$

Substitute back for $u$:

$\frac{1}{2} (6 + \sin(x))^8 + C$

Final Answers:

• A: $6 \sin(9 + 11x^2) + C$
• B: $\frac{1}{2} (6 + \sin(x))^8 + C$

Would you like more details or help with anything else? Here are some related questions you might find useful:

1. What other substitution techniques are useful in integration?
2. Can we apply integration by parts to these integrals?
3. How can we check the correctness of these indefinite integrals?
4. How do we handle definite integrals with the same techniques?
5. What are common mistakes in using substitution for integrals?

Tip: Remember to always carefully check the differential in u-substitution. If it's off by a factor, the whole integral will not work correctly!