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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

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