Use integration by substitution and the Fundamental Theorem to evaluate the definite integral....

Question:

Use integration by substitution and the Fundamental Theorem to evaluate the definite integral. Enter the exact answer.

{eq}\displaystyle \int_0^2 11 x (x^2 + 1)^2\ dx {/eq}.

Definite Integral:

Definite Integrals are those integrals which do contain upper and lower limit. There is no constant of integration in the definite integral. To evaluate the definite integral, first, integrate it simply considering as indefinite integral and then apply the limits to find out the final result. To make integration easier any function can be substitute as a new variable say ?u?

Reverse power rule of integration is used to find out the integral of the form {eq}x^n. {/eq} It can be given by the following formula:

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

where C is the constant of integration.

Answer and Explanation:

Given:

{eq}\displaystyle \int_0^2 11 x (x^2 + 1)^2\ dx \\ {/eq}

On substituing {eq}u = (x^2 + 1), {/eq} we get:

On differentiating the above equation, we get:

{eq}du = 2xdx \\ {/eq}

On putting these values int he standard intagral, we get:

{eq}\int \frac {11}{2} u^2 \ du \\ {/eq}

On integrating the above integral,we get:

{eq}= \left [ \frac {11}{2} \frac {u^3}{ 3} \right ]^2_0 \\ = \left [ \frac {11}{6} {u^3} \right ]^2_0 \\ {/eq}

On putting these values in the above expression, we get:

{eq}= \frac {11}{6} \left [ (x^2 + 1)^3 \right ]^2_0 \\ {/eq}

On applying the limits to the above expression, we get:

{eq}= \frac {11}{6} \left [ ( (2^2 + 1)^3 ) - ( (0^2 + 1)^3) \right ] \\ = \frac {11}{6} \left [ ( 125 ) - ( 1) \right ] \\ = \frac {11}{6} \left [ 124 \right ] \\ = 227.33 \\ {/eq}

which is the answer.


Learn more about this topic:

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How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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