Mathematics, 18.03.2021 02:10 murtaghliam1

Find the integral of 1\1+(2x)^2

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Answer from: destinyhammons12345

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

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right

Equality Properties

Multiplication Property of EqualityDivision Property of EqualityAddition Property of EqualitySubtraction Property of Equality

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

f(x) = cxⁿf’(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]: $\displaystyle \int {\frac{1}{a^2 + u^2}} \, du = \frac{1}{a}arctan(\frac{u}{a}) + C$

Step-by-step explanation:

Step 1: Define

$\displaystyle \int {\frac{1}{1+(2x)^2}} \, dx$

Step 2: Integrate Pt. 1

[Integral] Rewrite: $\displaystyle \int {\frac{1}{1^2+(2x)^2}} \, dx$

Step 3: Identify Variables

Identify variables for u-substitution of arctrig.

Set u: $\displaystyle u = 2x$ Differentiate [Basic Power Rule]: $\displaystyle \frac{du}{dx} = 1 \cdot 2x^{1 - 1}$ [Derivative] Simplify: $\displaystyle \frac{du}{dx} = 2$ [Derivative] Rewrite: $\displaystyle du = 2dx$ Set a: $\displaystyle a = 1$

Step 4: Integrate Pt. 2

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

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

Unit: Integration of Arctrig

Book: College Calculus 10e

Answer from: Quest

the answer is true.

step-by-step explanation:

Answer from: Quest

answer: translate the circles so they share a common center point and dilate circle y by a scale factor of 3.

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