Evaluate the integral. int(log_(9)x)/(x)dx

Convert to Natural Logarithm: Let's use the change of base formula for logarithms to convert log base 9 to a natural logarithm (ln), which is more convenient for integration. log_9(x) = ln(x)/ln(9) Now, rewrite the integral using this conversion. ∫(log_9(x)/x)dx = ∫(ln(x)/(x*ln(9)))dxFactor Out Constant: Since ln(9) is a constant, we can factor it out of the integral. ∫(ln(x)/(x*ln(9)))dx = (1/ln(9))∫(ln(x)/x)dxIntegrate with Respect to x: Now, we recognize that the integral ∫(ln(x)/x)dx is a standard integral that equals (ln(x))^2/2. So, we integrate (ln(x)/x) with respect to x. (1/ln(9))∫(ln(x)/x)dx = (1/ln(9)) * (ln(x))^2/2 + CFinal Answer: We have found the indefinite integral. The final answer is (ln(x))^2/(2*ln(9)) + C.

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Find indefinite integrals using the substitution

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Q. Evaluate the integral.

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\int \frac{\log_{9}x}{x}dx

Convert to Natural Logarithm: Let's use the change of base formula for logarithms to convert $\log_{9}(x)$ to a natural logarithm ( $\ln$ ), which is more convenient for integration. $\newline$ $\log_{9}(x) = \frac{\ln(x)}{\ln(9)}$ $\newline$ Now, rewrite the integral using this conversion. $\newline$ $\int\left(\frac{\log_{9}(x)}{x}\right)dx = \int\left(\frac{\ln(x)}{x\cdot\ln(9)}\right)dx$
Factor Out Constant: Since $\ln(9)$ is a constant, we can factor it out of the integral. $\int\left(\frac{\ln(x)}{x\cdot\ln(9)}\right)dx = \left(\frac{1}{\ln(9)}\right)\int\left(\frac{\ln(x)}{x}\right)dx$
Integrate with Respect to x: Now, we recognize that the integral $\int(\ln(x)/x)\,dx$ is a standard integral that equals $(\ln(x))^2/2$ . So, we integrate $(\ln(x)/x)$ with respect to $x$ . $(1/\ln(9))\int(\ln(x)/x)\,dx = (1/\ln(9)) \cdot (\ln(x))^2/2 + C$
Final Answer: We have found the indefinite integral. $\newline$ The final answer is $\frac{(\ln(x))^2}{2\cdot\ln(9)} + C$ .