91Ó°ÊÓ

Evaluate the following integrals in problems 1 to \(20 .\) No calculators are allowed. (However, you may use calculators to check your results.) $$\int \frac{1}{x \ln x} d x$$

Evaluate the following integrals in problems 1 to \(20 .\) No calculators are allowed. (However, you may use calculators to check your results.) $$\int \ln \left(e^{5 x+1}\right) d x$$

Evaluate the following integrals in problems 1 to \(20 .\) No calculators are allowed. (However, you may use calculators to check your results.) $$\int \frac{e^{4 x}-1}{e^{x}} d x$$

Evaluate the following integrals in problems 1 to \(20 .\) No calculators are allowed. (However, you may use calculators to check your results.) $$\int\left(9-x^{2}\right) \sqrt{x} d x$$

Evaluate the following integrals in problems 1 to \(20 .\) No calculators are allowed. (However, you may use calculators to check your results.) $$\int \sqrt{x}\left(1+x^{3 / 2}\right)^{4} d x$$

Evaluate the following integrals in problems 1 to \(20 .\) No calculators are allowed. (However, you may use calculators to check your results.) If \(\frac{d y}{d x}=e^{x}+2\) and the point (0,6) is on the graph of \(y,\) find \(y .\)

Evaluate the following integrals in problems 1 to \(20 .\) No calculators are allowed. (However, you may use calculators to check your results.) $$\int-3 e^{x} \sin \left(e^{x}\right) d x$$

Evaluate the following integrals in problems 1 to \(20 .\) No calculators are allowed. (However, you may use calculators to check your results.) $$\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x$$

If \(f(x)\) is the antiderivative of \(\frac{1}{x}\) and \(f(1)=5,\) find \(f(e)\).

Air is pumped into a spherical balloon, whose maximum radius is 10 meters. For what value of \(r\) is the rate of increase of the volume a hundred times that of the radius?