Question

Find the integral of 1\1+(2x)^2​

Answer 1

`\displaystyle \int {(1)/(1+(2x)^2)} \, dx = (artan(2x))/(2) + C`

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Equality Properties

• Multiplication Property of Equality
• Division Property of Equality
• Addition Property of Equality
• Subtraction Property of Equality

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Antiderivatives - Integrals

Indefinite Integrals

• Integral Constant C

U-Substitution

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

Arctrig Integration [arctangent]:
`\displaystyle \int {(1)/(a^2 + u^2)} \, du = (1)/(a)arctan((u)/(a)) + C`

Explanation:

Step 1: Define

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

Step 2: Integrate Pt. 1

1. [Integral] Rewrite:
   `\displaystyle \int {(1)/(1^2+(2x)^2)} \, dx`

Step 3: Identify Variables

Identify variables for u-substitution of arctrig.

1. Set u:
   `\displaystyle u = 2x`
2. Differentiate [Basic Power Rule]:
   `\displaystyle (du)/(dx) = 1 \cdot 2x^(1 - 1)`
3. [Derivative] Simplify:
   `\displaystyle (du)/(dx) = 2`
4. [Derivative] Rewrite:
   `\displaystyle du = 2dx`
5. Set a:
   `\displaystyle a = 1`

Step 4: Integrate Pt. 2

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle (1)/(2)\int {(2)/(1^2+(2x)^2)} \, dx`
2. [Integral] U-Substitution:
   `\displaystyle (1)/(2)\int {(du)/(a +u^2)}`
3. [Integral] Arctrig Integration [arctangent]:
   `\displaystyle (1)/(2)[(1)/(a)arctan((u)/(a))] + C`
4. [Integral] Back-Substitute:
   `\displaystyle (1)/(2)[(1)/(1)arctan((2x)/(1))] + C`
5. [Integral] Divide:
   `\displaystyle (1)/(2)[arctan(2x)] + C`
6. [Integral] Multiply:
   `\displaystyle (arctan(2x))/(2) + C`

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

Unit: Integration of Arctrig

Book: College Calculus 10e