Question

Find the integral of x(4x² + 1) from 0 to 2. a. 18 c. 22 b. 16 d. 20

Answer 1

a. 18

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

Distributive Property

Algebra I

• Terms/Coefficients

Calculus

Integrals

• Definite Integrals

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

Integration Rule [Fundamental Theorem of Calculus 1]:
`\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)`

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`

Explanation:

Step 1: Define

Identify

`\displaystyle \int\limits^2_0 {x(4x^2 + 1)} \, dx`

Step 2: Integrate

1. [Integrand] Distribute x [Distributive Property]:
   `\displaystyle \int\limits^2_0 {(4x^3 + x)} \, dx`
2. Rewrite Integral [Integration Property - Addition/Subtraction]:
   `\displaystyle \int\limits^2_0 {4x^3} \, dx + \int\limits^2_0 {x} \, dx`
3. Rewrite 1st Integral [Integration Property - Multiplied Constant]:
   `\displaystyle 4\int\limits^2_0 {x^3} \, dx + \int\limits^2_0 {x} \, dx`
4. [Integrals] Reverse Power Rule:
   `\displaystyle 4((x^4)/(4)) \bigg| \limits^2_0 + ((x^2)/(2)) \bigg| \limits^2_0`
5. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   `\displaystyle 4(4) + 2`
6. Evaluate:
   `\displaystyle 18`

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

Unit: Integration

Book: College Calculus 10e