Solve: int(ln(x)+5)/(x)dx

Recognize Split Integral: Recognize that the integral can be split into two separate integrals. I = ∫(ln(x) + 5)/x dx I = ∫ln(x)/x dx + ∫5/x dxIntegrate ln(x)/x: Integrate the first part ∫ln(x)/x dx using the fact that the integral of ln(x)/x is a known integral. I1 = ∫ln(x)/x dx = (ln(x))^2/2Integrate 5/x: Integrate the second part ∫5/x dx using the fact that the integral of 1/x is ln|x|. I2 = ∫5/x dx = 5ln|x|Combine Final Integral: Combine the results from Step 2 and Step 3 to get the final indefinite integral. I = I1 + I2 + C I = (ln(x))^2/2 + 5ln|x| + C

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Find indefinite integrals using the substitution and by parts

Full solution

Q. Solve:

\int \frac{\ln(x) + 5}{x} \, dx

Recognize Split Integral: Recognize that the integral can be split into two separate integrals. $\newline$ $I = \int(\ln(x) + 5)/x \, dx$ $\newline$ $I = \int\ln(x)/x \, dx + \int5/x \, dx$
Integrate $\ln(x)/x$ : Integrate the first part $\int \ln(x)/x \, dx$ using the fact that the integral of $\ln(x)/x$ is a known integral. $\newline$ $I_1 = \int \ln(x)/x \, dx = (\ln(x))^2/2$
Integrate $5/x$ : Integrate the second part $\int \frac{5}{x} \, dx$ using the fact that the integral of $1/x$ is $\ln|x|$ . $I_2 = \int \frac{5}{x} \, dx = 5\ln|x|$
Combine Final Integral: Combine the results from Step $2$ and Step $3$ to get the final indefinite integral. $\newline$ $I = I_1 + I_2 + C$ $\newline$ $I = \frac{(\ln(x))^2}{2} + 5\ln|x| + C$