Question

Exercise is mixed —some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration.

∫x^2√x+4 dx.

Answer 1

`\displaystyle \int {x^2√(x + 4)} \, dx = \frac{2(x + 4)^\Big{(3)/(2)}(15x^2 - 48x + 128)}{105} + C`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals
• [Indefinite Integrals] Integration Constant C

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (x^(n + 1))/(n + 1) + C`

Integration Property [Multiplied Constant]:
`\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx`

Integration Property [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

U-Substitution

• U-Solve

Explanation:

Step 1: Define

Identify

`\displaystyle \int {x^2√(x + 4)} \, dx`

Step 2: Integrate Pt. 1

Identify variables for u-solve.

1. Set u:
   `\displaystyle u = x + 4`
2. [u] Rewrite:
   `\displaystyle x = u - 4`
3. [u] Manipulate:
   `\displaystyle x^2 = (u - 4)^2`
4. [u] Basic Power Rule:
   `\displaystyle du = dx`

Step 3: Integrate Pt. 2

1. [Integral] U-Solve:
   `\displaystyle \int {x^2√(x + 4)} \, dx = \int {(u - 4)^2√(u)} \, dx`
2. [Integrand] Rewrite:
   `\displaystyle \int {x^2√(x + 4)} \, dx = \int {u^\Big{(5)/(2)} - 8u^\Big{(3)/(2)} + 16u^\Big{(1)/(2)}} \, dx`
3. [Integral] Rewrite [Integration Property - Addition/Subtraction]:
   `\displaystyle \int {x^2√(x + 4)} \, dx = \int {u^\Big{(5)/(2)}} \, dx - \int {8u^\Big{(3)/(2)}} \, dx + \int {16u^\Big{(1)/(2)}} \, dx`
4. [Integrals] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {x^2√(x + 4)} \, dx = \int {u^\Big{(5)/(2)}} \, dx - 8 \int {u^\Big{(3)/(2)}} \, dx + 16 \int {u^\Big{(1)/(2)}} \, dx`
5. [Integrals] Integration Rule [Reverse Power Rule]:
   `\displaystyle \int {x^2√(x + 4)} \, dx = \frac{2u^\Big{(7)/(2)}}{7} - 8 \Bigg( \frac{2u^\Big{(5)/(2)}}{5} \Bigg) + 16 \Bigg( \frac{2u^\Big{(3)/(2)}}{3} \Bigg) + C`
6. Simplify:
   `\displaystyle \int {x^2√(x + 4)} \, dx = \frac{2u^\Big{(7)/(2)}}{7} - \frac{16u^\Big{(5)/(2)}}{5} + \frac{32u^\Big{(3)/(2)}}{3} + C`
7. [u] Back-Substitute:
   `\displaystyle \int {x^2√(x + 4)} \, dx = \frac{2(x + 4)^\Big{(7)/(2)}}{7} - \frac{16(x + 4)^\Big{(5)/(2)}}{5} + \frac{32(x + 4)^\Big{(3)/(2)}}{3} + C`
8. Rewrite:
   `\displaystyle \int {x^2√(x + 4)} \, dx = \frac{2(x + 4)^\Big{(3)/(2)}(15x^2 - 48x + 128)}{105} + C`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration