Calculus

# Integration of Logarithmic Functions

If $$xf'(x)=24\ln x,$$ what is the function $$f(x)?$$ (Use $$C$$ as the constant of integration.)

If $$f(x)$$ is a function such that $f(x)=\int \frac{1}{x \sqrt{\ln x+3}} dx$ and $$f(e)=21,$$ what is the value of $$f(e^{5})?$$

What is the indefinite integral$\int \frac{1}{x}\left(7(\ln x)^{6}+5\right) dx?$ (Use $$C$$ as the constant of integration.)

Let $$N = \displaystyle \int_{1}^{e^ 8} \ln \left( \dfrac {1}{x} \right) \, dx$$. If $$N = -ae^b - c$$, where $$a$$, $$b$$ and $$c$$ are positive integers and $$e$$ is the base of the natural logarithm, what is the value of $$a + b + c$$?

If $$f(x)$$ is a function satisfying $f(x)=\int \frac{6(\ln x)^{5}}{x} dx$ and $$f(e)=8,$$ what is the value of $$f(e^2)?$$

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