Question

Find the general indefinite integral. (use c for the constant of integration.) (7x2 + 8x−2) dx

Answer 1

`\displaystyle \int ({7x^2 + 8x - 2)} \, dx = (7x^3)/(3) + 4x^2 - 2x + C`

General Formulas and Concepts:

Calculus

Integration

• Integrals
• [Indefinite Integrals] Integration Constant C

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

Explanation:

Step 1: Define

Identify

`\displaystyle \int ({7x^2 + 8x - 2)} \, dx`

Step 2: Integrate

1. [Integral] Rewrite [Integration Property - Addition/Subtraction]:
   `\displaystyle \int ({7x^2 + 8x - 2)} \, dx = \int {7x^2} \, dx + \int {8x} \, dx - \int {2} \, dx`
2. [Integrals] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int ({7x^2 + 8x - 2)} \, dx = 7\int {x^2} \, dx + 8\int {x} \, dx - 2\int {} \, dx`
3. [Integrals] Reverse Power Rule:
   `\displaystyle \int ({7x^2 + 8x - 2)} \, dx = 7 \Big( (x^3)/(3) \Big)+ 8 \Big( (x^2)/(2) \Big) - 2x + C`
4. Simplify:
   `\displaystyle \int ({7x^2 + 8x - 2)} \, dx = (7x^3)/(3) + 4x^2 - 2x + C`

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

Unit: Integration