int_((1)/(e))^(e)(tan^(-1)x)/(x)dx

Identify Integral: Identify the integral to be solved. We need to evaluate the integral of the function (tan^(-1)x)/(x) within the limits from 1/e to e.Check Standard Formula: Check if there is a standard integration formula that can be applied. The integral of (tan^(-1)x)/(x) does not match any standard integration formula directly. Therefore, we will need to use integration techniques to solve it.Apply Integration by Parts: Apply integration by parts. Integration by parts is given by the formula ∫u dv = uv - ∫v du, where u and dv are parts of the integrand. We choose u = tan^(-1)x (which gives du = (1/(1+x^2))dx) and dv = dx/x (which gives v = ln|x|).Perform Integration: Perform the integration by parts. Using the formula from Step 3, we have: ∫(tan^(-1)x)/(x) dx = tan^(-1)x * ln|x| - ∫ln|x| * (1/(1+x^2)) dxEvaluate Remaining Integral: Evaluate the remaining integral. The integral ∫ln|x| * (1/(1+x^2)) dx is not elementary and cannot be expressed in terms of elementary functions. This suggests that there might be a mistake in the choice of u and dv in the integration by parts.

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$\int_{\frac{1}{e}}^{e} \frac{\tan^{-1}x}{x}dx$

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Evaluate definite integrals using the chain rule

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\int_{\frac{1}{e}}^{e} \frac{\tan^{-1}x}{x}dx

Identify Integral: Identify the integral to be solved. $\newline$ We need to evaluate the integral of the function $\frac{\tan^{-1}x}{x}$ within the limits from $\frac{1}{e}$ to $e$ .
Check Standard Formula: Check if there is a standard integration formula that can be applied. $\newline$ The integral of $(\tan^{-1}x)/(x)$ does not match any standard integration formula directly. Therefore, we will need to use integration techniques to solve it.
Apply Integration by Parts: Apply integration by parts. Integration by parts is given by the formula $\int u \, dv = uv - \int v \, du$ , where $u$ and $dv$ are parts of the integrand. We choose $u = \tan^{-1}x$ (which gives $du = \frac{1}{1+x^2}$ dx) and $dv = \frac{dx}{x}$ (which gives $v = \ln|x|$ ).
Perform Integration: Perform the integration by parts. $\newline$ Using the formula from Step $3$ , we have: $\newline$ $\int\frac{\tan^{-1}x}{x} dx = \tan^{-1}x \cdot \ln|x| - \int\ln|x| \cdot \frac{1}{1+x^2} dx$
Evaluate Remaining Integral: Evaluate the remaining integral. $\newline$ The integral $\int \ln|x| \cdot \left(\frac{1}{1+x^2}\right) dx$ is not elementary and cannot be expressed in terms of elementary functions. This suggests that there might be a mistake in the choice of $u$ and $dv$ in the integration by parts.