Evaluate the integral \int_{2}^{5} x^3 dx

Question:

Evaluate the integral

{eq}\int_{2}^{5} x^3 dx {/eq}

Power Rule For Integrals

The power rule for derivatives is the following:

{eq}(x^n)' = n x^{n-1} {/eq}

Therefore, the power rule for integrals is :

{eq}\int x^n \; dx = \frac{1}{n+1} x^{n+1} {/eq}

Answer and Explanation:

According to power rule for integrals:

{eq}I = \int_2^5 x^3 \; dx = \frac{1}{4} \left [ x^{4} \right ]_2^5\\ {/eq}

Evaluating the limits of integration we get:

{eq}I = \frac{600-16}{4} \\ = \frac{584}{4} \\ = 146 {/eq}


Learn more about this topic:

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Evaluating Definite Integrals Using the Fundamental Theorem

from AP Calculus AB: Exam Prep

Chapter 16 / Lesson 2
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