Question

Find the following integral.
Integrate x + 5/x^2 +10x +26 dx

Answer 1

`\displaystyle \int {(x + 5)/(x^2 + 10x + 26)} \, dx = (ln|x^2 + 10x + 26|)/(2) + C`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals
• Definite/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`

U-Substitution

Explanation:

Step 1: Define

Identify

`\displaystyle \int {(x + 5)/(x^2 + 10x + 26)} \, dx`

Step 2: Integrate Pt. 1

Set variables for u-substitution.

1. Set u:
   `\displaystyle u = x^2 + 10x + 26`
2. [u] Differentiate [Addition/Subtraction, Basic Power Rule]:
   `\displaystyle du = 2x + 10 \ dx`

Step 3: Integrate Pt. 2

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {(x + 5)/(x^2 + 10x + 26)} \, dx = (1)/(2)\int {(2(x + 5))/(x^2 + 10x + 26)} \, dx`
2. [Integrand] Expand:
   `\displaystyle \int {(x + 5)/(x^2 + 10x + 26)} \, dx = (1)/(2)\int {(2x + 10)/(x^2 + 10x + 26)} \, dx`
3. [Integral] U-Substitution:
   `\displaystyle \int {(x + 5)/(x^2 + 10x + 26)} \, dx = (1)/(2)\int {(1)/(u)} \, du`
4. [Integral] Logarithmic Integration:
   `\displaystyle \int {(x + 5)/(x^2 + 10x + 26)} \, dx = (1)/(2)ln|u| + C`
5. Back-Substitute:
   `\displaystyle \int {(x + 5)/(x^2 + 10x + 26)} \, dx = (1)/(2)ln|x^2 + 10x + 26| + C`
6. Simplify:
   `\displaystyle \int {(x + 5)/(x^2 + 10x + 26)} \, dx = (ln|x^2 + 10x + 26|)/(2) + C`

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

Unit: Integration