Question

How to solve the indefinite integral

Answer 1

`(1)/(4)x^4-x^2+3x+c`

Explanation:

Use the the integration rules for powers of x:
`\int x^n dx = (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 = \\=(1)/(4)x^4-x^2+3x+c`

with c being an arbitrary constant.

Answer 2

`\displaystyle f(x) = (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 = (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 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) = (x^4)/(4) - 2 \bigg( (x^2)/(2) \bigg) + 3x + C`
4. Simplify:
   `\displaystyle f(x) = (x^4)/(4) - x^2 + 3x + C`

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

Unit: Integration