Problem 65
Solve the initial-value problem for \(x\) as a function of \(t\). \((t+5) \frac{d x}{d t}=x^{2}+1, t>-5, x(1)=\tan 1\)
Problem 65
For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible. \(\int_{-\pi / 3}^{\pi / 3} \sqrt{\sec ^{2} x-1} d x\)
Problem 65
This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of \(F\) is the set of all real numbers s such that the improper integral converges. Find the Laplace transform \(F\) of each of the following functions and give the domain of \(F\).\(f(x)=x\)
Problem 65
Find the area of the region enclosed by the curve \(y=x \cos x\) and the \(x\) -axis for \(\frac{11 \pi}{2} \leq x \leq \frac{13 \pi}{2} .\) (Express the answer in exact form.)
Problem 66
Find the volume of the solid generated by revolving the region bounded by the curve \(y=\ln x\), the \(x\) -axis, and the vertical line \(x=e^{2}\) about the \(x\) -axis. (Express the answer in exact form.)
Problem 66
This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of \(F\) is the set of all real numbers s such that the improper integral converges. Find the Laplace transform \(F\) of each of the following functions and give the domain of \(F\).\(f(x)=\cos (2 x)\)
Problem 66
For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible. \(\int_{0}^{\pi / 2} \sqrt{1-\cos (2 x)} d x\)
Problem 66
Solve the initial-value problem for \(x\) as a function of \(t\). \(\left(2 t^{3}-2 t^{2}+t-1\right) \frac{d x}{d t}=3, x(2)=0\)
Problem 67
Find the area of the region bounded by the graphs of the equations \(y=\sin x, y=\sin ^{3} x, x=0\), and \(x=\frac{\pi}{2}\).
Problem 67
The error formula for Simpson's rule depends on a. \(f(x)\) b. \(f^{\prime}(x)\) c. \(f^{(4)}(x)\) d. the number of steps
