Chapter 5: Q. 86 (page 465)

Solve given definite integral.
$\int_{0}^{4} x^{3} \sqrt{x^{2} + 4} d x$

Short Answer

Expert verified

242.78

Step by step solution

Step1. Given Information

The integral is as follows.

$\int_{0}^{4} x^{3} \sqrt{x^{2} + 4} d x$

The objective is to solve the integral.

Step2. Assumptions

To solve the integral, let $u = x^{2}$

Now, derivation of above equation is solved below.

$u = x^{2} d u = 2 x d x$

Step3. Solution

The integral is solved below.

$\begin{aligned} \int x^{3} \sqrt{x^{2} + 4} d x \\ = \int \frac{u}{2} \sqrt{u + 4} d u \\ = \frac{1}{2} \int u \sqrt{u + 4} d u \end{aligned}$

$= \frac{1}{2} \int 2 v^{2} (v^{2} - 4) d v [v = \sqrt{u + 4}, d v = \frac{9}{2 \sqrt{u + 4}} d u]$

$\begin{aligned} = \int v^{4} - 4 v^{2} d v \\ = \int v^{4} d v - 4 \int v^{2} d v \\ = \frac{1}{5} (v)^{5} - \frac{4}{3} (v)^{3} \end{aligned}$

$= \frac{1}{5} {(x^{2} + 4)}^{\frac{5}{2}} - \frac{4}{3} {(x^{2} + 4)}^{\frac{3}{2}}$

$= (\frac{x^{2}}{5} - \frac{8}{15}) {(x^{2} + 4)}^{\frac{3}{2}}$

Step4. Solution

The definite integral is solved below.

$\int_{0}^{4} x^{3} \sqrt{x^{2} + 4} d x$

$= {[(\frac{x^{2}}{5} - \frac{8}{15}) {(x^{2} + 4)}^{\frac{3}{2}}]}_{0}^{4}$

$= [(\frac{4^{2}}{5} - \frac{8}{15}) {(4^{2} + 4)}^{\frac{3}{2}}] - [(\frac{0^{2}}{5} - \frac{8}{15}) {(0^{2} + 4)}^{\frac{3}{2}}]$

$= \frac{320 \sqrt{5}}{3} + \frac{64}{15}$

$= 242.78$

Therefore, the value is $242.78$

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Solve given definite integral.
$\int_{0}^{4} x^{3} \sqrt{x^{2} + 4} d x$

Short Answer

Step by step solution

Step1. Given Information

Step2. Assumptions

Step3. Solution

Step4. Solution

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Solve given definite integral.∫04 x3x2+4dx

Short Answer

Step by step solution

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Step3. Solution

Step4. Solution

One App. One Place for Learning.

Most popular questions from this chapter

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Solve given definite integral.
$\int_{0}^{4} x^{3} \sqrt{x^{2} + 4} d x$