Answered

Expand your knowledge base with the help of IDNLearn.com's extensive answer archive. Explore a wide array of topics and find reliable answers from our experienced community members.

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

Sagot :

Spaceidn

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

We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for trusting IDNLearn.com with your questions. Visit us again for clear, concise, and accurate answers.