Question

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

Answer 1

Answer:

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

General Formulas and Concepts:

Calculus

Integration

• Integrals
• [Indefinite Integrals] Integration Constant C

Integration Rule [Reverse Power Rule]:                                                               $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

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:

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:                                                                   $\displaystyle \int ({7x^2 + 8x - 2)} \, dx = 7 \Big( \frac{x^3}{3} \Big)+ 8 \Big( \frac{x^2}{2} \Big) - 2x + C$
4. Simplify:                                                                                                         $\displaystyle \int ({7x^2 + 8x - 2)} \, dx = \frac{7x^3}{3} + 4x^2 - 2x + C$  

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

Unit: Integration