Question

how to solve the indefinite integral

Answer 1

Answer:

$\frac{1}{4}x^4-x^2+3x+c$

Step-by-step explanation:

Use the the integration rules for powers of x: $\int x^n dx = \frac{1}{(n+1)} x^{n+1}$ and the additive property of integrals:

$f(x) = \int (x^3-2x+3)dx= \int x^3 dx-2\int x dx+\int 3 dx = \\=\frac{1}{4}x^4-x^2+3x+c$

with c being an arbitrary constant.

Answer 2

Answer:

$\displaystyle f(x) = \frac{x^4}{4} - x^2 + 3x + C$

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

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

Step 2: Integrate

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

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

Unit: Integration