Evaluate the integral. \int \sqrt[3]{x} dx a. (3/4)x^(4/3) + C b. (3/2)x^(2/3) + C c....


Evaluate the integral.

{eq}\displaystyle \int \sqrt[3]{x}\, dx {/eq}


a. {eq}\; \left(\frac{3}{4}\right)x^{(4/3)} + C {/eq}

b. {eq}\; \left(\frac{3}{2}\right)x^{(2/3)} + C {/eq}

c. {eq}\; \frac{1}{3x^{(2/3)}} + C {/eq}

d. {eq}\; \textrm{None of these.} {/eq}

Rules of Integration:

Integration means antiderivative. But it is not possible to evaluate every integral based on this definition. A set of rules is developed based on this definition to evaluate the integrals. One of them is:

$$\int x^n \, dx = \dfrac{x^{n+1}}{n+1}+C $$

Answer and Explanation:

The given integral is:

$$\begin{align} \int \sqrt[3]{x}\, dx & = \int x^{1/3} dx \\ &= \dfrac{x^{1/3+1}}{1/3+1} +C & (\because \int x^n \, dx = \dfrac{x^{n+1}}{n+1}+C) \\ &= \dfrac{x^{4/3}}{(4/3)}+C \\ &= \dfrac{3}{4} x^{4/3}+C \end{align} $$

Therefore, the answer is option (a).

Learn more about this topic:

Integration Problems in Calculus: Solutions & Examples

from AP Calculus AB & BC: Homework Help Resource

Chapter 13 / Lesson 13

Related to this Question

Explore our homework questions and answers library