Problem 17
In Exercises \(11-20,\) solve the initial value problem explicitly. \(\frac{d y}{d t}=\frac{1}{1+t^{2}}+2^{t} \ln 2\) and \(y=3\) when \(t=0\)
Problem 18
In Exercises \(17-24,\) use the indicated substitution to evaluate the integral. Confirm your answer by differentiation. $$\int x \cos \left(2 x^{2}\right) d x, \quad u=2 x^{2}$$
Problem 18
In Exercises \(15-18,\) solve the differential equation. $$G^{\prime}(t)=\frac{2 t^{3}}{t^{3}-t}$$
Problem 18
In Exercises \(11-20,\) solve the initial value problem explicitly. \(\frac{d x}{d t}=\frac{1}{t}-\frac{1}{t^{2}}+6\) and \(x=0\) when \(t=1\)
Problem 18
In Exercises \(17-20,\) use parts and solve for the unknown integral. $$\int e^{-x} \cos x d x$$
Problem 19
\(\int \frac{2 x}{x^{2}-4} d x\)
Problem 19
In Exercises \(17-24,\) use the indicated substitution to evaluate the integral. Confirm your answer by differentiation. $$\int \sec 2 x \tan 2 x d x, u=2 x$$
Problem 19
In Exercises \(17-20,\) use parts and solve for the unknown integral. $$\int e^{x} \cos 2 x d x$$
Problem 19
In Exercises \(11-20,\) solve the initial value problem explicitly. \(\frac{d v}{d t}=4 \sec t \tan t+e^{t}+6 t\) and \(v=5\) when \(t=0\)
Problem 20
In Exercises 19 and \(20,\) find the amount of time required for a \(\$ 2000\) investment to double if the annual interest rate \(r\) is compounded (a) annually, (b) monthly, (c) quarterly, and (d) continuously. \(r=8.25 \%\)
