Answer:
\(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = 2arctan(x) + \frac{ln|x^2 + 1|}{2} + C\)
General Formulas and Concepts:
Algebra I
- Terms/Coefficients
- Expanding
Calculus
Differentiation
- Derivatives
- Derivative Notation
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:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Integration
- Integrals
- [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]: \(\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx\)
Integration Property [Addition/Subtraction]: \(\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx\)
U-Substitution
Step-by-step explanation:
Step 1: Define
Identify
\(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx\)
Step 2: Integrate Pt. 1
Set variables for u-substitution.
- Set u: \(\displaystyle u = x^2 + 1\)
- [u] Differentiate [Basic Power Rule, Addition/Subtraction]: \(\displaystyle du = 2x \ dx\)
Step 3: Integrate Pt. 2
- [Integral] Rewrite [Integration Property - Multiplied Constant]: \(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2}\int {\frac{2(x + 2)}{x^2 + 1}} \, dx\)
- [Integrand] Expand: \(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2}\int {\frac{2x + 4}{x^2 + 1}} \, dx\)
- [Integral] Rewrite [Integration Property - Addition/Subtraction]: \(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ \int {\frac{2x}{x^2 + 1}} \, dx + \int {\frac{4}{x^2 + 1}} \, dx \bigg]\)
- [2nd Integral] Rewrite [Integration Property - Multiplied Constant]: \(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ \int {\frac{2x}{x^2 + 1}} \, dx + 4\int {\frac{1}{x^2 + 1}} \, dx \bigg]\)
- [1st Integral] U-Substitution: \(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ \int {\frac{1}{u}} \, du + 4\int {\frac{1}{x^2 + 1}} \, dx \bigg]\)
- [1st Integral] Logarithmic Integration: \(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ ln|u| + 4\int {\frac{1}{x^2 + 1}} \, dx \bigg]\)
- [Integral] Arctrig Integration: \(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ ln|u| + 4 \bigg( \frac{1}{1}arctan \Big( \frac{x}{1} \Big) \bigg) \bigg] + C\)
- Simplify: \(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = \frac{1}{2} \bigg[ ln|u| + 4arctan(x) \bigg] + C\)
- Expand: \(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = 2arctan(x) + \frac{ln|u|}{2} + C\)
- Back-Substitute: \(\displaystyle \int {\frac{x + 2}{x^2 + 1}} \, dx = 2arctan(x) + \frac{ln|x^2 + 1|}{2} + C\)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
