Question

find the indefinit integral x^3-1/x^2 dx​

Answer 1

Answer:

$\displaystyle \int {\frac{x^3 - 1}{x^2}} \, dx = \frac{x^2}{2} + \frac{1}{x} + C$

General Formulas and Concepts:

Algebra I

• Exponential Property [Dividing]:                                                                   $\displaystyle \frac{b^m}{b^n} = b^{m - n}$
• Exponential Property [Rewrite]:                                                                   $\displaystyle b^{-m} = \frac{1}{b^m}$

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

$\displaystyle \int {\frac{x^3 - 1}{x^2}} \, dx$

Step 2: Integrate

1. [Integrand] Rewrite:                                                                                       $\displaystyle \int {\frac{x^3 - 1}{x^2}} \, dx = \int {\bigg( \frac{x^3}{x^2} - \frac{1}{x^2} \bigg)} \, dx$
2. Simplify [Exponential Property - Dividing]:                                                  $\displaystyle \int {\frac{x^3 - 1}{x^2}} \, dx = \int {\bigg( x - \frac{1}{x^2} \bigg)} \, dx$
3. [Integral] Rewrite [Integration Property - Addition/Subtraction]:               $\displaystyle \int {\frac{x^3 - 1}{x^2}} \, dx = \int {x} \, dx - \int {\frac{1}{x^2}} \, dx$
4. [2nd Integral] Rewrite [Exponential Property - Rewrite]:                             $\displaystyle \int {\frac{x^3 - 1}{x^2}} \, dx = \int {x} \, dx - \int {x^{-2}} \, dx$
5. [Integrals] Reverse Power Rule:                                                                   $\displaystyle \int {\frac{x^3 - 1}{x^2}} \, dx = \frac{x^2}{2} - (-x^{-1}) + C$
6. Simplify/Rewrite [Exponential Property - Rewrite]:                                     $\displaystyle \int {\frac{x^3 - 1}{x^2}} \, dx = \frac{x^2}{2} + \frac{1}{x} + C$

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

Unit: Integration