Q3P Express the following integrals ... [FREE SOLUTION]

Chapter 11: Q3P (page 544)

Express the following integrals as $尾$ functions, and then, by (7.1), in terms of $螕$ functions. When possible, use $螕$ function formulas to write an exact answer in terms of $蟺, 2$ , etc. Compare your answers with computer results and reconcile any discrepancies.
3. ${鈭�}_{0}^{1} \frac{1}{\sqrt{1 - x^{3}}} d x$ .

Short Answer

Expert verified

The value of integral is $I = \frac{1}{3} 尾 (\frac{1}{3}, \frac{1}{2})$ and $I = \frac{1}{3} 脳 \frac{螕 (\frac{1}{3}) 螕 (\frac{1}{2})}{螕 (\frac{5}{6})}$ .

Step by step solution

Given Information

The given Integral is ${鈭�}_{0}^{1} \frac{1}{\sqrt{1 - x^{3}}}$ .

Definition of Beta function

Beta function is defined as $尾 (p, q) = {鈭�}_{0}^{1} t^{p - 1} {(1 - t)}^{q - 1} d t$ .

Express integral as Beta integral

The Integral is ${鈭�}_{0}^{1} \frac{1}{\sqrt{1 - x^{3}}}$ .

Formula for beta function is $尾 (p, q) = {鈭�}_{0}^{1} t^{p - 1} {(1 - t)}^{q - 1} d t$ .

Let $x^{3} = y a$ and find its derivate with respect to x.

$\begin{matrix} x^{3} = y \\ 3 x^{2} d x = d y \\ d x = \frac{1}{3 y^{\frac{2}{3}}} d y \end{matrix}$

Substitute the value of $x^{3}$ in the integral.

The equation becomes as follows.

$\begin{array}{l} I = {鈭�}_{0}^{1} \frac{1}{\sqrt{1 - y}} \frac{1}{3 y^{\frac{2}{3}}} d y \\ I = {鈭�}_{0}^{1} \frac{y^{\frac{- 2}{3}}}{\sqrt{1 - y}} \frac{1}{3} d y \\ I = \frac{1}{3} 尾 (\frac{1}{3}, \frac{1}{2}) \end{array}$

Express integral as gamma integral.

The relation between $尾$ and $螕$ is the equation mentioned below.

$尾 (m, n) = \frac{螕 (m) 螕 (n)}{螕 (m + n)}$

Substitute the values given below in the equation mentioned above.

$\begin{matrix} m = \frac{1}{3} \\ n = \frac{1}{2} \end{matrix}$

The equation becomes as follows.

$\begin{array}{l} 尾 (\frac{1}{3}, \frac{1}{2}) = \frac{螕 (\frac{1}{3}) 螕 (\frac{1}{2})}{螕 (\frac{1}{3} + \frac{1}{2})} \\ 尾 (\frac{1}{3}, \frac{1}{2}) = \frac{螕 (\frac{1}{3}) 螕 (\frac{1}{2})}{螕 (\frac{5}{6})} \\ I = \frac{1}{3} 脳 \frac{螕 (\frac{1}{3}) 螕 (\frac{1}{2})}{螕 (\frac{5}{6})} \end{array}$

Hence, the value of integral is $I = \frac{1}{3} 尾 (\frac{1}{3}, \frac{1}{2})$ and $I = \frac{1}{3} 尾 (\frac{1}{3}, \frac{1}{2})$ .

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Short Answer

Step by step solution

Given Information

Definition of Beta function

Express integral as Beta integral

Express integral as gamma integral.

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91影视

Express the following integrals as 尾functions, and then, by (7.1), in terms of 螕 functions. When possible, use 螕 function formulas to write an exact answer in terms of 蟺,2, etc. Compare your answers with computer results and reconcile any discrepancies.3. 鈭�0111-x3dx.

Short Answer

Step by step solution

Given Information

Definition of Beta function

Express integral as Beta integral

Express integral as gamma integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

Electromagnetism

Linear Momentum

Electrostatics

Oscillations

Wave Optics

Study anywhere. Anytime. Across all devices.