Anti-Derivative Worksheet II - Answers: A table listing 17 calculus problems with their corresponding anti-derivative solutions.

The provided image is an "Anti-Derivative Worksheet II – Answers" that lists various integrals and their corresponding antiderivatives. Below, I will explain the solution process for a few selected problems to illustrate the methods used in solving these integrals.

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Problem 1:

$$
\int (x-2)^3 \, dx
$$

#### Solution:
To solve this integral, we use the substitution method. Let:
$$
u = x - 2 \quad \Rightarrow \quad du = dx
$$
Substituting \( u \) into the integral:
$$
\int (x-2)^3 \, dx = \int u^3 \, du
$$
Now, integrate \( u^3 \):
$$
\int u^3 \, du = \frac{u^4}{4} + C
$$
Substitute back \( u = x - 2 \):
$$
\frac{u^4}{4} + C = \frac{(x-2)^4}{4} + C
$$
Thus, the answer is:
$$
\boxed{y = \frac{1}{4}(x-2)^4 + C}
$$

---

Problem 4:

$$
\int \frac{2}{\sqrt{x+1}} \, dx
$$

#### Solution:
We use the substitution method. Let:
$$
u = x + 1 \quad \Rightarrow \quad du = dx
$$
Substituting \( u \) into the integral:
$$
\int \frac{2}{\sqrt{x+1}} \, dx = \int \frac{2}{\sqrt{u}} \, du
$$
Rewrite \( \frac{1}{\sqrt{u}} \) as \( u^{-1/2} \):
$$
\int \frac{2}{\sqrt{u}} \, du = \int 2u^{-1/2} \, du
$$
Now, integrate \( 2u^{-1/2} \):
$$
\int 2u^{-1/2} \, du = 2 \cdot \frac{u^{1/2}}{\frac{1}{2}} + C = 2 \cdot 2u^{1/2} + C = 4u^{1/2} + C
$$
Substitute back \( u = x + 1 \):
$$
4u^{1/2} + C = 4\sqrt{x+1} + C
$$
Thus, the answer is:
$$
\boxed{y = 4\sqrt{x+1} + C}
$$

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Problem 9:

$$
\int 3x^2(x^3 - 1) \, dx
$$

#### Solution:
We use the substitution method. Let:
$$
u = x^3 - 1 \quad \Rightarrow \quad du = 3x^2 \, dx
$$
Notice that \( 3x^2 \, dx \) is already present in the integral, so we can directly substitute:
$$
\int 3x^2(x^3 - 1) \, dx = \int u \, du
$$
Now, integrate \( u \):
$$
\int u \, du = \frac{u^2}{2} + C
$$
Substitute back \( u = x^3 - 1 \):
$$
\frac{u^2}{2} + C = \frac{(x^3 - 1)^2}{2} + C
$$
Thus, the answer is:
$$
\boxed{y = \frac{1}{2}(x^3 - 1)^2 + C}
$$

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Problem 15:

$$
\int 3x\sqrt{1-x^2} \, dx
$$

#### Solution:
We use the substitution method. Let:
$$
u = 1 - x^2 \quad \Rightarrow \quad du = -2x \, dx \quad \Rightarrow \quad x \, dx = -\frac{1}{2} \, du
$$
Substitute \( u \) and \( x \, dx \) into the integral:
$$
\int 3x\sqrt{1-x^2} \, dx = \int 3 \cdot \sqrt{u} \cdot \left(-\frac{1}{2}\right) \, du = -\frac{3}{2} \int \sqrt{u} \, du
$$
Rewrite \( \sqrt{u} \) as \( u^{1/2} \):
$$
-\frac{3}{2} \int \sqrt{u} \, du = -\frac{3}{2} \int u^{1/2} \, du
$$
Now, integrate \( u^{1/2} \):
$$
-\frac{3}{2} \int u^{1/2} \, du = -\frac{3}{2} \cdot \frac{u^{3/2}}{\frac{3}{2}} + C = -u^{3/2} + C
$$
Substitute back \( u = 1 - x^2 \):
$$
-u^{3/2} + C = -(1 - x^2)^{3/2} + C
$$
Thus, the answer is:
$$
\boxed{y = -(1 - x^2)^{3/2} + C}
$$

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General Explanation:

The solutions to these integrals primarily rely on the following techniques:
1. Substitution Method: Replacing a complicated expression with a new variable \( u \) to simplify the integral.
2. Power Rule: For integrals of the form \( \int u^n \, du \), the power rule states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) (for \( n \neq -1 \)).
3. Chain Rule Application: When dealing with composite functions, the substitution helps "undo" the chain rule during differentiation.

Each problem in the worksheet follows a similar approach, applying these fundamental techniques to find the antiderivative.

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Final Answer:

The answers provided in the worksheet are correct. The detailed solutions above demonstrate the methods used to arrive at those answers. If you have any specific questions about other problems, feel free to ask!

$$
\boxed{\text{All answers in the worksheet are correct.}}
$$