Evaluate. (k lambda)/(4R)int_(-(pi)/(2))^(pi//2)sec ((theta)/(2))*d theta

Identify Integral: Identify the integral that needs to be evaluated. We have the integral ∫ sec(θ/2) dθ with limits from -(π/2) to π/2.Simplify Integrands: Use a trigonometric identity to simplify the integrand. The secant function can be expressed as sec(x) = 1/cos(x). Therefore, sec(θ/2) = 1/cos(θ/2).Recognize Integral Rule: Recognize that the integral of sec(x) is ln|sec(x) + tan(x)| + C. However, since we have sec(θ/2), we need to adjust the integral accordingly.Perform Substitution: Perform a substitution to integrate sec(θ/2). Let u = θ/2, which implies dθ = 2du. We need to change the limits of integration as well. When θ = -(π/2), u = -(π/4), and when θ = π/2, u = π/4.Rewrite in Terms of u: Rewrite the integral in terms of u. The integral becomes ∫ sec(u) * 2du with limits from -(π/4) to π/4.Evaluate with New Limits: Evaluate the integral with the new limits. ∫ sec(u) * 2du from -(π/4) to π/4 = 2 * ln|sec(u) + tan(u)| from -(π/4) to π/4.Calculate Definite Integral: Calculate the definite integral. 2 * [ln|sec(π/4) + tan(π/4)| - ln|sec(-(π/4)) + tan(-(π/4))|].Simplify Using Trig Functions: Simplify the expression using the fact that sec(π/4) = √2 and tan(π/4) = 1 (and similarly for -(π/4)). 2 * [ln(√2 + 1) - ln(√2 - 1)].Combine Logarithmic Terms: Use the properties of logarithms to combine the terms. 2 * ln[(√2 + 1)/(√2 - 1)].Rationalize Denominator: Rationalize the denominator of the argument of the logarithm. 2 * ln[((√2 + 1)(√2 + 1))/((√2 - 1)(√2 + 1))] = 2 * ln[(2 + 2√2 + 1)/(2 - 1)] = 2 * ln(3 + 2√2).Multiply by Constant Factor: Multiply the result by the constant factor (k lambda)/(4R). (k lambda)/(4R) * 2 * ln(3 + 2√2) = (k lambda)/(2R) * ln(3 + 2√2).

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Calculus

Evaluate definite integrals using the power rule

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Q. Evaluate.

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\frac{k \lambda}{4 R} \int_{-\frac{\pi}{2}}^{\pi / 2} \sec \frac{\theta}{2} \cdot d \theta

Identify Integral: Identify the integral that needs to be evaluated. $\newline$ We have the integral $\int_{-\left(\frac{\pi}{2}\right)}^{\frac{\pi}{2}} \sec\left(\frac{\theta}{2}\right) d\theta$ with limits from $-\left(\frac{\pi}{2}\right)$ to $\frac{\pi}{2}$ .
Simplify Integrands: Use a trigonometric identity to simplify the integrand. The secant function can be expressed as $\sec(x) = \frac{1}{\cos(x)}$ . Therefore, $\sec(\frac{\theta}{2}) = \frac{1}{\cos(\frac{\theta}{2})}$ .
Recognize Integral Rule: Recognize that the integral of $\sec(x)$ is $\ln|\sec(x) + \tan(x)| + C$ . However, since we have $\sec(\theta/2)$ , we need to adjust the integral accordingly.
Perform Substitution: Perform a substitution to integrate $\sec(\frac{\theta}{2})$ . Let $u = \frac{\theta}{2}$ , which implies $d\theta = 2du$ . We need to change the limits of integration as well. When $\theta = -\left(\frac{\pi}{2}\right)$ , $u = -\left(\frac{\pi}{4}\right)$ , and when $\theta = \frac{\pi}{2}$ , $u = \frac{\pi}{4}$ .
Rewrite in Terms of u: Rewrite the integral in terms of u. $\newline$ The integral becomes $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sec(u) \cdot 2\,du$ with limits from $-\frac{\pi}{4}$ to $\frac{\pi}{4}$ .
Evaluate with New Limits: Evaluate the integral with the new limits. $\newline$ $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \sec(u) \cdot 2\,du = 2 \cdot \ln|\sec(u) + \tan(u)|$ from $-\frac{\pi}{4}$ to $\frac{\pi}{4}$ .
Calculate Definite Integral: Calculate the definite integral. $\newline$ $2 \times [\ln|\sec(\frac{\pi}{4}) + \tan(\frac{\pi}{4})| - \ln|\sec(-\frac{\pi}{4}) + \tan(-\frac{\pi}{4})|]$ .
Simplify Using Trig Functions: Simplify the expression using the fact that $\sec(\frac{\pi}{4}) = \sqrt{2}$ and $\tan(\frac{\pi}{4}) = 1$ (and similarly for $-\frac{\pi}{4}$ ). $\newline$ $2 \cdot [\ln(\sqrt{2} + 1) - \ln(\sqrt{2} - 1)]$ .
Combine Logarithmic Terms: Use the properties of logarithms to combine the terms. $2 \times \ln\left[\frac{(\sqrt{2} + 1)}{(\sqrt{2} - 1)}\right].$
Rationalize Denominator: Rationalize the denominator of the argument of the logarithm. $\newline$ $2 \cdot \ln\left(\frac{(\sqrt{2} + 1)(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)}\right) = 2 \cdot \ln\left(\frac{2 + 2\sqrt{2} + 1}{2 - 1}\right) = 2 \cdot \ln(3 + 2\sqrt{2})$ .
Multiply by Constant Factor: Multiply the result by the constant factor $(\frac{k \lambda}{4R})$ . $\frac{k \lambda}{4R} \times 2 \times \ln(3 + 2\sqrt{2}) = \frac{k \lambda}{2R} \times \ln(3 + 2\sqrt{2})$ .