Use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition.
$\dfrac{d y}{d x}=\dfrac{10}{x \sqrt{x^2-1}}$

$$y(3)=0$$

Find the indefinite integral. \

$$\displaystyle\int \sin ^3 x \cos x d x$$

Determine whether the function $f(x)=\frac{1}{x-4}$ is integrable on the interval $[3,5]$. Explain.

Find the indefinite integral. \

$$\displaystyle\int e^x\left(e^x+1\right)^2 d x$$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the given interval. Sketch the region. \

$$y=16-x^2, \quad[1,3]$$

Find F prime of x. \

$$\displaystyle F(x)=\int_1^x \frac{1}{t} d t$$

A differential equation, a point, and a slope field are given.
Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in the previous part \

$$\dfrac{d y}{d x}=\dfrac{2}{\sqrt{25-x^2}}, \quad(5, \pi)$$

A differential equation, a point, and a slope field are given.
ketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. \

$$\dfrac{d y}{d x}=\dfrac{2}{\sqrt{25-x^2}}, \quad(5, \pi)$$

Find the indefinite integral. \

$$\displaystyle\int \csc ^2\left(\frac{x}{2}\right) d x$$

Find the integral.
$\displaystyle\int \frac{\cosh x}{\sqrt{9-\sinh ^2 x}} d x$

Use the limit process to find the area of the region between the graph of the function and the x-axis over the given interval. Sketch the region.
$y=x^2+1, \quad[0,3]$

Find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. \

$$f(x)=\dfrac{4\left(x^2+1\right)}{x^2},[1,3]$$

Solve the differential equation. \

$$f^{\prime \prime}(x)=x^2, f^{\prime}(0)=6, f(0)=3$$

Use a computer algebra system to find or evaluate the integral. \

$$\displaystyle\int_{-\pi / 4}^{\pi / 4} \frac{\sin ^2 x-\cos ^2 x}{\cos x} d x$$