Mathematics worksheet on antiderivatives from Al Falahiyya School, UAE, for the 2020/2021 academic year.

Problem: Solve the given antiderivative problems and explain the solution.

#### Q1: Find the antiderivative of \( \int (4x^2 - 3) \, dx \)

Solution:
To find the antiderivative, we integrate each term separately:
1. The antiderivative of \( 4x^2 \):
\[
\int 4x^2 \, dx = 4 \int x^2 \, dx = 4 \cdot \frac{x^3}{3} = \frac{4}{3}x^3
\]
2. The antiderivative of \( -3 \):
\[
\int -3 \, dx = -3x
\]

Combining these results, we get:
\[
\int (4x^2 - 3) \, dx = \frac{4}{3}x^3 - 3x + C
\]

Correct Answer:
\[
\boxed{d) \, \frac{4}{3}x^3 - 3x + C}
\]

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#### Q2: Find the antiderivative of \( \int \sec^2 x \, dx \)

Solution:
The integral of \( \sec^2 x \) is a standard result:
\[
\int \sec^2 x \, dx = \tan x + C
\]

Correct Answer:
\[
\boxed{b) \, \tan x + C}
\]

---

#### Q3: Find the antiderivative of \( \int \frac{4}{x^2} \, dx \)

Solution:
Rewrite \( \frac{4}{x^2} \) as \( 4x^{-2} \):
\[
\int \frac{4}{x^2} \, dx = \int 4x^{-2} \, dx
\]
Now, integrate using the power rule:
\[
\int 4x^{-2} \, dx = 4 \int x^{-2} \, dx = 4 \cdot \frac{x^{-1}}{-1} = 4 \cdot \left( -\frac{1}{x} \right) = -\frac{4}{x} + C
\]

Correct Answer:
\[
\boxed{d) \, -\frac{4}{x} + C}
\]

---

#### Q4: Find the antiderivative of \( \int 3\sqrt{x} \, dx \)

Solution:
Rewrite \( \sqrt{x} \) as \( x^{1/2} \):
\[
\int 3\sqrt{x} \, dx = \int 3x^{1/2} \, dx
\]
Now, integrate using the power rule:
\[
\int 3x^{1/2} \, dx = 3 \int x^{1/2} \, dx = 3 \cdot \frac{x^{3/2}}{3/2} = 3 \cdot \frac{2}{3} x^{3/2} = 2x^{3/2} + C
\]

Correct Answer:
\[
\boxed{a) \, 2x^{3/2} + C}
\]

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#### Q5: Find the antiderivative of \( \int \frac{4}{\sqrt{1-x^2}} \, dx \)

Solution:
The integral \( \int \frac{1}{\sqrt{1-x^2}} \, dx \) is a standard result:
\[
\int \frac{1}{\sqrt{1-x^2}} \, dx = \sin^{-1} x + C
\]
Thus, for \( \int \frac{4}{\sqrt{1-x^2}} \, dx \), we can factor out the constant 4:
\[
\int \frac{4}{\sqrt{1-x^2}} \, dx = 4 \int \frac{1}{\sqrt{1-x^2}} \, dx = 4 \sin^{-1} x + C
\]

Correct Answer:
\[
\boxed{b) \, 4\sin^{-1} x + C}
\]

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Final Answers:

1. \( \boxed{d) \, \frac{4}{3}x^3 - 3x + C} \)
2. \( \boxed{b) \, \tan x + C} \)
3. \( \boxed{d) \, -\frac{4}{x} + C} \)
4. \( \boxed{a) \, 2x^{3/2} + C} \)
5. \( \boxed{b) \, 4\sin^{-1} x + C} \)