Calculus
Antiderivatives
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 \(f(e^2)?\)