Question

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

Answer 1

Answer:

$\displaystyle \int {\frac{x + 5}{x^2 + 10x + 26}} \, dx = \frac{ln|x^2 + 10x + 26|}{2} + C$

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

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]:                                                               $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

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

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int {\frac{x + 5}{x^2 + 10x + 26}} \, dx$

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

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

Unit:  Integration