Evaluate int_(1)^(3)(7^(ln x))/(x)dx

Recognize Inverse Relationship: Recognize that the natural logarithm function is the inverse of the exponential function. We have the integral: int_(1)^(3)(7^(ln x))/(x)dx Notice that 7^(ln x) can be rewritten using properties of exponents and logarithms as x^(ln 7), since a^(ln b) = b^(ln a). So, the integral becomes: int_(1)^(3)x^(ln 7)/x dxRewrite Integral with Exponents: Simplify the integrand. The x in the denominator cancels out one x from the numerator, simplifying the integral to: int_(1)^(3)x^(ln 7 - 1) dx Now, the integral is: int_(1)^(3)x^(ln 7 - 1) dxSimplify Integrands: Integrate using the power rule for integration. The power rule states that the integral of x^n is (x^(n+1))/(n+1) + C, where n ≠ -1. Applying the power rule, we get: int_(1)^(3)x^(ln 7 - 1) dx = [(x^(ln 7))/ln 7] from 1 to 3Apply Power Rule: Evaluate the definite integral. We need to evaluate the expression at the upper and lower limits of the integral and subtract: [(3^(ln 7))/ln 7] - [(1^(ln 7))/ln 7] Since any number to the power of 0 is 1, 1^(ln 7) is 1. So, the evaluation becomes: [(3^(ln 7))/ln 7] - [1/ln 7]Evaluate Definite Integral: Calculate the final answer. We can now compute the values: [(3^(ln 7))/ln 7] - [1/ln 7] = (3^(ln 7) - 1)/ln 7

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Evaluate $\int_{1}^{3} \frac{7^{\ln x}}{x} \, dx$

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Calculus

Evaluate definite integrals using the chain rule

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Q. Evaluate

\int_{1}^{3} \frac{7^{\ln x}}{x} \, dx

Recognize Inverse Relationship: Recognize that the natural logarithm function is the inverse of the exponential function. $\newline$ We have the integral: $\newline$ $\int_{1}^{3}\frac{7^{\ln x}}{x}\,dx$ $\newline$ Notice that $7^{\ln x}$ can be rewritten using properties of exponents and logarithms as $x^{\ln 7}$ , since $a^{\ln b} = b^{\ln a}$ . $\newline$ So, the integral becomes: $\newline$ $\int_{1}^{3}\frac{x^{\ln 7}}{x}\,dx$
Rewrite Integral with Exponents: Simplify the integrand. $\newline$ The $x$ in the denominator cancels out one $x$ from the numerator, simplifying the integral to: $\newline$ $\int_{1}^{3}x^{\ln 7 - 1} \, dx$ $\newline$ Now, the integral is: $\newline$ $\int_{1}^{3}x^{\ln 7 - 1} \, dx$
Simplify Integrands: Integrate using the power rule for integration. $\newline$ The power rule states that the integral of $x^n$ is $(x^{(n+1)})/(n+1) + C$ , where $n \neq -1$ . $\newline$ Applying the power rule, we get: $\newline$ $\int_{1}^{3}x^{(\ln 7 - 1)} dx = \left[\frac{x^{(\ln 7)}}{\ln 7}\right]$ from $1$ to $3$
Apply Power Rule: Evaluate the definite integral. $\newline$ We need to evaluate the expression at the upper and lower limits of the integral and subtract: $\newline$ $[\frac{3^{(\ln 7)}}{\ln 7}] - [\frac{1^{(\ln 7)}}{\ln 7}]$ $\newline$ Since any number to the power of $0$ is $1$ , $1^{(\ln 7)}$ is $1$ . $\newline$ So, the evaluation becomes: $\newline$ $[\frac{3^{(\ln 7)}}{\ln 7}] - [\frac{1}{\ln 7}]$
Evaluate Definite Integral: Calculate the final answer. $\newline$ We can now compute the values: $\newline$ $[\frac{3^{(\ln 7)}}{\ln 7}] - [\frac{1}{\ln 7}] = \frac{3^{(\ln 7)} - 1}{\ln 7}$