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Determine the following indefinite integrals. Check your work by differentiation.

∫4x^1/3-x^-1/3dx

Sagot :

Space

Answer:

[tex]\displaystyle \int { \Big( 4x^\bigg{\frac{1}{3}} - x^\bigg{\frac{-1}{3}} \Big) } \, dx = 3x^\bigg{\frac{4}{3}} - \frac{3x^\bigg{\frac{2}{3}}}{2} + C[/tex]

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                           [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

Derivative Property [Addition/Subtraction]:                                                         [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals
  • Indefinite Integrals
  • Integration Constant C

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

Integration Property [Multiplied Constant]:                                                         [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Property [Addition/Subtraction]:                                                       [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \int { \Big( 4x^\bigg{\frac{1}{3}} - x^\bigg{\frac{-1}{3}} \Big) } \, dx[/tex]

Step 2: Integrate

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

Step 3: Check

Differentiate the answer.

  1. Rewrite [Derivative Property - Addition/Subtraction]:                                 [tex]\displaystyle \frac{d}{dx} \bigg[ 3x^\bigg{\frac{4}{3}} \bigg] - \frac{d}{dx} \bigg[ \frac{3x^\bigg{\frac{2}{3}}}{2} \bigg] + \frac{d}{dx} \bigg[ C \bigg][/tex]
  2. Rewrite [Derivative Property - Multiplied Constant]:                                   [tex]\displaystyle 3\frac{d}{dx} \bigg[ x^\bigg{\frac{4}{3}} \bigg] - \frac{3}{2}\frac{d}{dx} \bigg[ x^\bigg{\frac{2}{3}} \bigg] + \frac{d}{dx} \bigg[ C \bigg][/tex]
  3. Basic Power Rule:                                                                                         [tex]\displaystyle 3 \bigg( \frac{4}{3}x^\bigg{\frac{1}{3}} \bigg) - \frac{3}{2} \bigg( \frac{2}{3}x^\bigg{\frac{-1}{3}} \bigg)[/tex]
  4. Simplify:                                                                                                         [tex]\displaystyle 4x^\bigg{\frac{1}{3}} - x^\bigg{\frac{-1}{3}}[/tex]

∴ we have found the answer.

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

Unit: Integration

Book: College Calculus 10e

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