Question

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

Answer 1

Answer:

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

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]:                                                               $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

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:

*Note:

Antiderivative = Integral

Step 1: Define

Identify.

$\displaystyle \int {x(2 - x)} \, dx$

Step 2: Integrate

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

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

Unit: Integration