Evaluate the integral:

{eq}\displaystyle \int\limits_{4}^{12} \sqrt {2x + 1}\ dx {/eq}.

Question:

Evaluate the integral:

{eq}\displaystyle \int\limits_{4}^{12} \sqrt {2x + 1}\ dx {/eq}.

Definite Integral:

To solve the given definite integral, first use the general property of the integral, that is

{eq}\int x^n \ dx = \frac{x^{n+1}}{n+1} {/eq} and then use the given limits to get the required value.

Answer and Explanation: 1

To find the value of the given integral

{eq}I = \displaystyle \int\limits_{4}^{12} \sqrt {2x + 1}\ dx {/eq}, use the general properties of integration.

Let {eq}2x + 1= u\\ \Rightarrow 2\ dx= du \\ \Rightarrow dx= \frac{1}{2}\ du {/eq}

Thus,

{eq}I = \frac{1}{2} \displaystyle \int\limits_{9}^{25} \sqrt {u}\ du\\ = \frac{1}{2} \displaystyle (u^{3/2})_{9}^{25}\\ = \frac{1}{2}\times \frac{2}{3} \displaystyle (25^{3/2}-9^{3/2})\\ = \frac{98}{3} {/eq}


Learn more about this topic:

Integral Calculus: Definition & Applications
Integral Calculus: Definition & Applications

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Chapter 12 / Lesson 2
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Understand that an integral measures the area under a curve, and learn how to evaluate linear and polynomial integrals. Explore different applications of integrals with examples.


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