Evaluate the integral. int_(0)^(7pi//4)tan ((x)/(7)dx)

Identify integral: Identify the integral to be solved. We have the integral: ∫(0 to 7π/4) tan(x/7) dxUse substitution: Use a substitution to simplify the integral. Let u = x/7, which implies that x = 7u. Then, we need to find dx in terms of du. dx = 7 duChange limits: Change the limits of integration according to the substitution. When x = 0, u = 0/7 = 0. When x = 7π/4, u = (7π/4)/7 = π/4. So the new limits of integration are from u = 0 to u = π/4.Rewrite in terms of u: Rewrite the integral in terms of u. ∫(0 to π/4) tan(u) * 7 duFactor out constant: Factor out the constant from the integral. 7 * ∫(0 to π/4) tan(u) duIntegrate tan(u): Integrate tan(u) with respect to u. The integral of tan(u) is -ln|cos(u)|. So we have: 7 * [-ln|cos(u)|] evaluated from 0 to π/4Evaluate at limits: Evaluate the antiderivative at the upper and lower limits. 7 * [-ln|cos(π/4)| + ln|cos(0)|] = 7 * [-ln(√2/2) + ln(1)] = 7 * [0 - (-ln(√2/2))] = 7 * ln(√2/2)Simplify expression: Simplify the expression. 7 * ln(√2/2) = 7 * ln(1/√2) = 7 * ln(2^(-1/2)) = 7 * (-1/2) * ln(2) = -7/2 * ln(2)

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Evaluate the integral. $\newline$ $\int_{0}^{\frac{7\pi}{4}} \tan \left( \frac{x}{7} \right) dx$

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Calculus

Evaluate definite integrals using the chain rule

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Q. Evaluate the integral.

\newline

\int_{0}^{\frac{7\pi}{4}} \tan \left( \frac{x}{7} \right) dx

Identify integral: Identify the integral to be solved. $\newline$ We have the integral: $\newline$ $\int_{0}^{7\pi/4} \tan(\frac{x}{7}) \, dx$
Use substitution: Use a substitution to simplify the integral. $\newline$ Let $u = \frac{x}{7}$ , which implies that $x = 7u$ . Then, we need to find $dx$ in terms of $du$ . $\newline$ $dx = 7 du$
Change limits: Change the limits of integration according to the substitution. $\newline$ When $x = 0$ , $u = \frac{0}{7} = 0$ . $\newline$ When $x = \frac{7\pi}{4}$ , $u = \frac{(7\pi/4)}{7} = \frac{\pi}{4}$ . $\newline$ So the new limits of integration are from $u = 0$ to $u = \frac{\pi}{4}$ .
Rewrite in terms of $u$ : Rewrite the integral in terms of $u$ . $\int_{0}^{\frac{\pi}{4}} \tan(u) \cdot 7 \, du$
Factor out constant: Factor out the constant from the integral. $7 \times \int_{0}^{\frac{\pi}{4}} \tan(u) \, du$
Integrate $\tan(u)$ : Integrate $\tan(u)$ with respect to $u$ . The integral of $\tan(u)$ is $-\ln|\cos(u)|$ . So we have: $7 \times [-\ln|\cos(u)|]$ evaluated from $0$ to $\frac{\pi}{4}$
Evaluate at limits: Evaluate the antiderivative at the upper and lower limits. $\newline$ $7 \times [-\ln|\cos(\pi/4)| + \ln|\cos(0)|]$ $\newline$ = $7 \times [-\ln(\sqrt{2}/2) + \ln(1)]$ $\newline$ = $7 \times [0 - (-\ln(\sqrt{2}/2))]$ $\newline$ = $7 \times \ln(\sqrt{2}/2)$
Simplify expression: Simplify the expression. $\newline$ $7 \cdot \ln(\sqrt{2}/2) = 7 \cdot \ln(1/\sqrt{2}) = 7 \cdot \ln(2^{(-1/2)}) = 7 \cdot (-1/2) \cdot \ln(2)$ $\newline$ $= -\frac{7}{2} \cdot \ln(2)$