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

Simplify Integrand: Simplify the integrand using properties of logarithms and exponents. We have the integral of (7^(ln(x)))/(x) from 1 to 3. We can use the property of exponents that a^(ln(b)) = b^(ln(a)) to simplify the integrand. Since ln(x) is the exponent to which the base 'e' must be raised to produce x, we can rewrite 7^(ln(x)) as x^(ln(7)). This gives us the new integrand x^(ln(7) - 1).Set Up Integral: Set up the integral with the simplified integrand. The integral becomes int_(1)^(3)x^(ln(7) - 1)dx. We can now integrate this expression with respect to x.Integrate Function: Integrate the function x^(ln(7) - 1) with respect to x. The antiderivative of x^n is (x^(n+1))/(n+1) + C, where n ≠ -1. In our case, n = ln(7) - 1, which is not equal to -1, so we can apply this rule. The antiderivative is (x^(ln(7)))/(ln(7)) + C.Evaluate Definite Integral: Evaluate the definite integral from 1 to 3. We substitute the limits of integration into the antiderivative to get: [(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. Therefore, the expression simplifies to: [(3^(ln(7)))/(ln(7))] - 1/(ln(7)).Calculate Integral Value: Calculate the value of the definite integral. We need to calculate 3^(ln(7)) which is the same as e^(ln(3)*ln(7)) since 3^(ln(7)) = e^(ln(3)*ln(7)). We then divide this by ln(7) and subtract 1/(ln(7)) to get the final answer. The final value is (e^(ln(3)*ln(7)))/(ln(7)) - 1/(ln(7)).

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$\int_{1}^{3}\left(\frac{7^{\ln(x)}}{x}\right)dx$

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Evaluate definite integrals using the chain rule

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\int_{1}^{3}\left(\frac{7^{\ln(x)}}{x}\right)dx

Simplify Integrand: Simplify the integrand using properties of logarithms and exponents. $\newline$ We have the integral of $(7^{\ln(x)})/(x)$ from $1$ to $3$ . We can use the property of exponents that $a^{\ln(b)} = b^{\ln(a)}$ to simplify the integrand. Since $\ln(x)$ is the exponent to which the base 'e' must be raised to produce $x$ , we can rewrite $7^{\ln(x)}$ as $x^{\ln(7)}$ . This gives us the new integrand $x^{(\ln(7) - 1)}$ .
Set Up Integral: Set up the integral with the simplified integrand. $\newline$ The integral becomes $\int_{1}^{3}x^{\ln(7) - 1}\,dx$ . We can now integrate this expression with respect to $x$ .
Integrate Function: Integrate the function $x^{(\ln(7) - 1)}$ with respect to $x$ . The antiderivative of $x^n$ is $\frac{x^{(n+1)}}{(n+1)} + C$ , where $n \neq -1$ . In our case, $n = \ln(7) - 1$ , which is not equal to $-1$ , so we can apply this rule. The antiderivative is $\frac{x^{(\ln(7))}}{(\ln(7))} + C$ .
Evaluate Definite Integral: Evaluate the definite integral from $1$ to $3$ . We substitute the limits of integration into the antiderivative to get: $\left(\frac{3^{(\ln(7))}}{\ln(7)}\right) - \left(\frac{1^{(\ln(7))}}{\ln(7)}\right)$ . Since any number to the power of $0$ is $1$ , $1^{(\ln(7))}$ is $1$ . Therefore, the expression simplifies to: $\left(\frac{3^{(\ln(7))}}{\ln(7)}\right) - \frac{1}{\ln(7)}$ .
Calculate Integral Value: Calculate the value of the definite integral. $\newline$ We need to calculate $3^{(\ln(7))}$ which is the same as $e^{(\ln(3)*\ln(7))}$ since $3^{(\ln(7))} = e^{(\ln(3)*\ln(7))}$ . We then divide this by $\ln(7)$ and subtract $\frac{1}{(\ln(7))}$ to get the final answer. $\newline$ The final value is $(\frac{e^{(\ln(3)*\ln(7))}}{\ln(7)}) - \frac{1}{\ln(7)}$ .