Question

Solve the Anti derivative.​

Answer 1

`\displaystyle \int {(1)/(9x^2+4)} \, dx = (1)/(6)arctan((3x)/(2)) + C`

General Formulas and Concepts:

Algebra I

• Factoring

Calculus

Antiderivatives - integrals/Integration

Integration Constant C

U-Substitution

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

Trig Integration:
`\displaystyle \int {(du)/(a^2 + u^2)} = (1)/(a)arctan((u)/(a)) + C`

Explanation:

Step 1: Define

`\displaystyle \int {(1)/(9x^2 + 4)} \, dx`

Step 2: Integrate Pt. 1

1. [Integral] Factor fraction denominator:
   `\displaystyle \int {(1)/(9(x^2 + (4)/(9)))} \, dx`
2. [Integral] Integration Property - Multiplied Constant:
   `\displaystyle (1)/(9) \int {(1)/(x^2 + (4)/(9))} \, dx`

Step 3: Identify Variables

Set up u-substitution for the arctan trig integration.

`\displaystyle u = x \\ a = (2)/(3) \\ du = dx`

Step 4: Integrate Pt. 2

1. [Integral] Substitute u-du:
   `\displaystyle (1)/(9) \int {(1)/(u^2 + ((2)/(3))^2) \, du`
2. [Integral] Trig Integration:
   `\displaystyle (1)/(9)[(1)/((2)/(3))arctan((u)/((2)/(3)))] + C`
3. [Integral] Simplify:
   `\displaystyle (1)/(9)[(3)/(2)arctan((3u)/(2))] + C`
4. [integral] Multiply:
   `\displaystyle (1)/(6)arctan((3u)/(2)) + C`
5. [Integral] Back-Substitute:
   `\displaystyle (1)/(6)arctan((3x)/(2)) + C`

Topic: AP Calculus AB

Unit: Integrals - Arctrig

Book: College Calculus 10e