int_(1)^(3)(1^(ln(x)))/(x)dx

Simplify the integrand: Simplify the integrand. The expression 1^(ln(x)) is equal to 1 for any value of x, because any number raised to the power of 0 is 1, and ln(x) is 0 when x=1. Therefore, the integrand simplifies to 1/x.Set up the integral: Set up the integral with the simplified integrand. The integral we need to evaluate is now ∫(1/x) dx from 1 to 3.Recall the antiderivative: Recall the antiderivative of 1/x. The antiderivative of 1/x is ln|x| + C, where C is the constant of integration.Evaluate the definite integral: Evaluate the definite integral. We need to find the value of ln|x| from 1 to 3. ∫(1/x) dx from 1 to 3 = ln|x| evaluated from 1 to 3 = ln(3) - ln(1)Calculate the final value: Calculate the final value. Since ln(1) is 0, the final value is ln(3) - 0, which simplifies to ln(3).

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$\int_{1}^{3}\left(1^{\ln(x)}\right)/x\,dx$

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Calculus

Evaluate definite integrals using the chain rule

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\int_{1}^{3}\left(1^{\ln(x)}\right)/x\,dx

Simplify the integrand: Simplify the integrand. $\newline$ The expression $1^{(\ln(x))}$ is equal to $1$ for any value of $x$ , because any number raised to the power of $0$ is $1$ , and $\ln(x)$ is $0$ when $x=1$ . Therefore, the integrand simplifies to $\frac{1}{x}$ .
Set up the integral: Set up the integral with the simplified integrand. $\newline$ The integral we need to evaluate is now $\int_{1}^{3}\frac{1}{x} \, dx$ .
Recall the antiderivative: Recall the antiderivative of $1/x$ . The antiderivative of $1/x$ is $ext{ln}|x| + C$ , where $C$ is the constant of integration.
Evaluate the definite integral: Evaluate the definite integral. $\newline$ We need to find the value of $\ln|x|$ from $1$ to $3$ . $\newline$ $\int_{1}^{3}(\frac{1}{x}) \, dx$ from $1$ to $3$ = $\ln|x|$ evaluated from $1$ to $3$ = $\ln(3)$ - $1$ $0$
Calculate the final value: Calculate the final value. $\newline$ Since $\ln(1)$ is $0$ , the final value is $\ln(3) - 0$ , which simplifies to $\ln(3)$ .