91Ó°ÊÓ

Evaluate each integral. $$\int \frac{d x}{x \sqrt{4-x^{8}}}$$

Evaluate each integral. $$\int \frac{d x}{x \sqrt{1+x^{4}}}$$

Evaluate the following integrals. Include absolute values only when needed. $$\int_{0}^{5} 5^{5 x} d x$$

General relative grow th rates Define the relative growth rate of the function \(f\) over the time interval \(T\) to be the relative change in f over an interval of length \(T\) : $$ R_{T}=\frac{f(t+T)-f(t)}{f(t)} $$ Show that for the exponential function \(y(t)=y_{0} e^{t},\) the relative growth rate \(R_{T}\), for fixed \(T\), is constant for all \(t\).

Evaluate the following integrals. Include absolute values only when needed. $$\int x^{2} 10^{x^{3}} d x$$

Evaluate each integral. $$\int \frac{\cosh z}{\sinh ^{2} z} d z$$

Equivalent growth functions The same exponential growth function can be written in the forms \(y(t)=y_{0} e^{t f}, y(t)=y_{0}(1+r)^{t}\) and \(y(t)=y_{0} 2^{1 / T_{2}}\). Write \(k\) as a function of \(r, r\) as a function of \(T_{2}\) and \(T_{2}\) as a function of \(k .\)

Evaluate the following integrals. Include absolute values only when needed. $$\int_{0}^{\pi} 2^{\sin x} \cos x d x$$

Geometric means A quantity grows exponentially according to \(y(t)=y_{0} e^{k t} .\) What is the relationship among \(m, n,\) and \(p\) such that \(y(p)=\sqrt{y(m) y(n)} ?\)

Evaluate each integral. $$\int \frac{\cos \theta}{9-\sin ^{2} \theta} d \theta$$