Estimate \int\limits_{1}^{2} 31x^{4} - 1 dx to within 1/100.


Estimate {eq}\int\limits_{1}^{2} 31x^{4} - 1 dx {/eq} to within 1/100.

Definite Integrals:

The definite integral given in the question is solved using the sum / difference rule of integration and the power rule of integration. TheSo the expansion is done using the sum / difference rule.The power rule will be as follows: {eq}\int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1 {/eq}

The rule sum / difference rule is {eq}\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx {/eq}

Answer and Explanation:

Given integral is :

{eq}\int _1^2\left(\:31x^4\:-\:1\right)\:dx\\ {/eq}

Evaluating the indefinite integral, we get:

{eq}=\int \:31x^4dx-\int \:1dx\\ =\frac{31x^5}{5}-x+C\\ {/eq}

Putting the limits of integration, we get;

{eq}\left [ \frac{31x^5}{5}-x \right ]_1^2\\ =\frac{982}{5}-\frac{26}{5}\\ =\frac{956}{5} {/eq}

Learn more about this topic:

Definite Integrals: Definition

from Math 104: Calculus

Chapter 10 / Lesson 6

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