In this given exercise, demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a).

$$F(x)=\int_{\pi / 4}^x \sec ^2 t d t$$

In this exercise, find the average value of the function over the given interval

$$f(x)=\frac{\ln x}{x}, \quad[1, e]$$

In this exercise, use the Trapezoidal Rule and Simpson's Rule with $n=4$, and use the integration capabilities of a graphing utility, to approximate the definite integral.

$$\int_1^2 \frac{1}{1+x^3} d x$$

Chemicals A and B combine in a 3-to-1 ratio to form a compound. The amount of compound $x$ being produced at any time $t$ is proportional to the unchanged amounts of $\mathrm{A}$ and $\mathrm{B}$ remaining in the solution. So, if 3 kilograms of $\mathrm{A}$ is mixed with 2 kilograms of $B$, you have

$$\frac{d x}{d t}=k\left(3-\frac{3 x}{4}\right)\left(2-\frac{x}{4}\right)=\frac{3 k}{16}\left(x^2-12 x+32\right) .$$

One kilogram of the compound is formed after 10 minutes. Find the amount formed after 20 minutes by solving the equation

$$\int \frac{3 k}{16} d t=\int \frac{d x}{x^2-12 x+32}$$

In this exercise, find the indefinite integral by the method shown in the earlier examples.

$$\int \frac{x^2-1}{\sqrt{2 x-1}} d x$$

In this given exercise, demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a).

$$F(x)=\int_4^x \sqrt{t} dt$$

In this exercise, consider a particle moving along the $x$-axis where $x(t)$ is the position of the particle at time $t, x^{\prime}(t)$ is its velocity, and $x^{\prime \prime}(t)$ is its acceleration.

A particle, initially at rest, moves along the $x$-axis such that its acceleration at time $t>0$ is given by $a(t)=\cos t$. At the time $t=0$, its position is $x=3$.

Find the values of $t$ for which the particle is at rest.

In this exercise, evaluate the integral in terms of inverse hyperbolic functions.

$$\int_{-1 / 2}^{1 / 2} \frac{d x}{1-x^2}$$

In this exercise, evaluate the integral in terms of natural logarithms.

$$\int_{-1 / 2}^{1 / 2} \frac{d x}{1-x^2}$$

In this given exercise, integrate to find $F$ as a function of $x$.

$$F(x)=\int_4^x \sqrt{t}dt$$

In this exercise, consider a particle moving along the $x$-axis where $x(t)$ is the position of the particle at time $t, x^{\prime}(t)$ is its velocity, and $x^{\prime \prime}(t)$ is its acceleration.

A particle, initially at rest, moves along the $x$-axis such that its acceleration at time $t>0$ is given by $a(t)=\cos t$. At the time $t=0$, its position is $x=3$.

Find the velocity and position functions for the particle.

After exercising for a few minutes, a person has a respiratory cycle for which the rate of air intake is $v=1.75 \sin (\pi t / 2)$. Find the volume, in liters, of air inhaled during one cycle by integrating the function over the interval $[0,2]$

In this exercise, find the indefinite integral by the method shown in the earlier examples.

$$\int(x+1) \sqrt{2-x} d x$$

In this exercise, use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of $n$. Round your answers to four decimal places and compare your results with the exact value of the definite integral.

$$\int_1^2 \frac{1}{(x+1)^2} d x, \quad n=4$$