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Chapter 11: Q12.4P (page 559)

$\int_{0}^{1} \frac{1}{\sqrt{1 - t^{2}} \sqrt{1 - \frac{t^{2}}{4}}} d t$

Short Answer

Expert verified

The value of integral in elliptic form is $F (\sin^{- 1} (\frac{1}{2}), \frac{1}{2})$ .

Step by step solution

01

Given Information

The value of integration is $\int_{0}^{1} \frac{1}{\sqrt{1 - t^{2}} \sqrt{1 - \frac{t^{2}}{4}}} d t$ .

02

Definition of elliptic form.

The elliptic form of the integral is defined as $F = \int_{0}^{1} \frac{1}{\sqrt{1 - t^{2}} \sqrt{1 - k^{2} t^{2}}} d t$ .

03

Find the value of Integral.

The value of integration is $\int_{0}^{1} \frac{1}{\sqrt{1 - t^{2}} \sqrt{1 - \frac{t^{2}}{4}}} d t$ .

The formula for the beta function is $F = \int_{0}^{1} \frac{1}{\sqrt{1 - t^{2}} \sqrt{1 - k^{2} t^{2}}} d t$ .

Equate the above equation with the value of I, the value of I becomes follows.

$l = \int_{0}^{1} \frac{1}{\sqrt{1 - t^{2}} \sqrt{1 - \frac{t^{2}}{4}}} d t l = F (\sin^{- 1} (\frac{1}{2}), \frac{1}{2})$

Hence, the value of integral in elliptic form is $l = F (\sin^{- 1} (\frac{1}{2}), \frac{1}{2})$ .

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Most popular questions from this chapter

In Chapter 1, equations (13.5) and (13.6), we defined the binomial coefficients $(p n)$ whereis a non-negative integer butmay be negative or fractional. Show that $(p n)$ can be written in terms of $Γ$ functions as

role="math" localid="1664340642097" $(\frac{p}{n}) = \frac{Γ (p + 1)}{n! Γ (p - n + 1)}$

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

13. $\int_{\frac{- 1}{2}}^{\frac{3}{4}} \frac{\sqrt{9 - 4 t^{2}}}{\sqrt{1 - t^{2}}} d t$

Verify equations (9.2), (9.3), and (9.4). Hint: In (9.2a) , you want to write $ϕ (x)$ in terms of an error function. Make the change of variable t = u √2 t = u (√2)in the $ϕ (x)$ integral. Warning: Don’t forget to adjust the limits; when $t = (x), u = \frac{x}{\sqrt{2}}$ .

In the pendulum problem, $θ = α s i n \sqrt{\frac{g}{l}} t$ is an approximate solution when the amplitude α is small enough for the motion to be considered simple harmonic. Show that the corresponding exact solution when α is not small is $s i n \frac{θ}{2} = s i n \frac{α}{2} s n \sqrt{\frac{g}{l}} t, k = s i n (\frac{α}{2})$ is the modulus of the elliptic function. Show that this reduces to the simple harmonic motion solution for small amplitude α

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

9. $\int_{\frac{- 1}{2}}^{\frac{1}{2}} \frac{d t}{\sqrt{1 - t^{2}} \sqrt{4 - 3 t^{2}}}$ .

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