Answered

Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

How to find antiderivative of x(2-x).

Sagot :

Space

Answer:

[tex]\displaystyle \int {x(2 - x)} \, dx = \frac{-x^2(x - 3)}{3} + C[/tex]

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

Algebra I

Terms/Coefficients

  • Factoring/Expanding

Calculus

Integration

  • Integrals
  • Integration Constant C

Integration Rule [Reverse Power Rule]:                                                               [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

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

Integration Property [Addition/Subtraction]:                                                       [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Step-by-step explanation:

*Note:

Antiderivative = Integral

Step 1: Define

Identify.

[tex]\displaystyle \int {x(2 - x)} \, dx[/tex]

Step 2: Integrate

  1. [Integrand] Expand:                                                                                       [tex]\displaystyle \int {x(2 - x)} \, dx = \int {2x - x^2} \, dx[/tex]
  2. [Integral] Rewrite [Integration Property - Addition/Subtraction]:               [tex]\displaystyle \int {x(2 - x)} \, dx = \int {2x} \, dx - \int {x^2} \, dx[/tex]
  3. [Left Integral] Rewrite [Integration Property - Multiplied Constant]:           [tex]\displaystyle \int {x(2 - x)} \, dx = 2 \int {x} \, dx - \int {x^2} \, dx[/tex]
  4. [Integrals] Integration Rule [Reverse Power Rule]:                                     [tex]\displaystyle \int {x(2 - x)} \, dx = 2 \bigg( \frac{x^2}{2} \bigg) - \frac{x^3}{3} + C[/tex]
  5. Simplify:                 ��                                                                                       [tex]\displaystyle \int {x(2 - x)} \, dx = x^2 - \frac{x^3}{3} + C[/tex]
  6. Factor:                                                                                                           [tex]\displaystyle \int {x(2 - x)} \, dx = x^2 \bigg( 1 - \frac{x}{3} \bigg) + C[/tex]
  7. Rewrite:                                                                                                         [tex]\displaystyle \int {x(2 - x)} \, dx = \frac{-x^2(x - 3)}{3} + C[/tex]

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

Unit: Integration

Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.