Mathematics, 27.03.2021 01:00 browneyedbaby20

What is the resulting integral?

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Answer from: sabrinaunderhillx

$\displaystyle \int {\frac{2}{(1 + \sqrt{x})^5}} \, dx = \frac{-(4\sqrt{x} + 1)}{3(\sqrt{x} + 1)^4} + C$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right

Algebra I

Exponential Rule [Root Rewrite]: $\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}$ Exponential Rule [Rewrite]: $\displaystyle b^{-m} = \frac{1}{b^m}$

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Antiderivatives - Integrals

Integration Constant C

U-Substitution

U-Solve

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]: $\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

Step-by-step explanation:

Step 1: Define

$\displaystyle \int {\frac{2}{(1 + \sqrt{x})^5}} \, dx$

Step 2: Identify Variables

Find the variables u-solve using u-substitution.

U-Substitution

$\displaystyle u = 1 + \sqrt{x}$

$\displaystyle du = \frac{1}{2\sqrt{x}}dx$

U-Solve

$\displaystyle x = (u - 1)^2$

$\displaystyle dx = 2\sqrt{x}du$

Step 3: Integration

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle 2\int {\frac{1}{(1 + \sqrt{x})^5}} \, dx$ [Integral] U-Solve: $\displaystyle 2\int {\frac{1}{(1 + \sqrt{(u - 1)^2})^5}} \, 2\sqrt{x}du$ [Integral] Simplify: $\displaystyle 2\int {\frac{2\sqrt{x}}{u^5}} \, du$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle 4\int {\frac{\sqrt{x}}{u^5}} \, du$ [Integral] U-Solve: $\displaystyle 4\int {\frac{\sqrt{(u - 1)^2}}{u^5}} \, du$ [Integral] Simplify: $\displaystyle 4\int {\frac{u - 1}{u^5}} \, du$ [Integral] Rewrite [Integration Property - Subtraction]: $\displaystyle 4[\int {\frac{u}{u^5}} \, du - \int {\frac{1}{u^5}} \, du]$ [Integrals] Simplify: $\displaystyle 4[\int {\frac{1}{u^4}} \, du - \int {\frac{1}{u^5}} \, du]$ [Integrals] Rewrite [Exponential Rule - Rewrite]: $\displaystyle 4[\int {u^{-4}} \, du - \int {u^{-5}} \, du]$ [Integrals] Reverse Power Rule: $\displaystyle 4[\frac{u^{-3}}{-3} - \frac{u^{-4}}{-4}] + C$ Rewrite [Exponential Rule - Rewrite]: $\displaystyle 4[\frac{-1}{3u^3} + \frac{1}{4u^4}] + C$ [Brackets] Multiply: $\displaystyle \frac{-4}{3u^3} + \frac{4}{4u^4} + C$ Back-Substitute: $\displaystyle \frac{-4}{3(1 + \sqrt{x})^3} + \frac{4}{4(1 + \sqrt{x})^4} + C$ Combine: $\displaystyle \frac{-(4\sqrt{x} + 1)}{3(\sqrt{x} + 1)^4} + C$

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

Unit: Integration

Book: College Calculus 10e

Answer from: Quest

nswer: a y-axis

step-by-step explanation:

Answer from: Quest

The second and fourth answer are correct

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