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

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

In this exercise, use the equation of the tractrix $y=a \operatorname{sech}^{-1} \frac{x}{a}-\sqrt{a^2-x^2}, \quad a>0$.

Let $L$ be the tangent line to the tractrix at the point $P$. If $L$ intersects the $y$-axis at the point $Q$, show that the distance between $P$ and $Q$ is $a$.

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

$$F(x)=\int_{-1}^x e^t d t$$

At the instant the traffic light turns green, a car that has been waiting at an intersection starts with a constant acceleration of 6 feet per second per second. At the same instant, a truck traveling with a constant velocity of 30 feet per second passes the car.

How fast will the car be traveling when it passes the truck?

At the instant the traffic light turns green, a car that has been waiting at an intersection starts with a constant acceleration of 6 feet per second per second. At the same instant, a truck traveling with a constant velocity of 30 feet per second passes the car.

How far beyond its starting point will the car pass the truck?

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

$$F(x)=\int_{-1}^x e^t d t$$

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

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

In this exercise, use the equation of the tractrix $y=a \operatorname{sech}^{-1} \frac{x}{a}-\sqrt{a^2-x^2}, \quad a>0$.

Find $d y / d x$.

A population of bacteria is changing at a rate of

$$\frac{d P}{d t}=\frac{3000}{1+0.25 t}$$

where $t$ is the time in days. The initial population (when $t=0$ ) is 1000 . Write an equation that gives the population at any time $t$, and find the population when $t=3$ days.

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.

$$\int_0^{\pi / 2} \sqrt{x} \cos x d x$$

An object is dropped from a height of 400 feet.

Use the result in part (c) to find $\lim _{t \rightarrow \infty} v(t)$ and give its interpretation.

An object is dropped from a height of 400 feet.

If the air resistance is proportional to the square of the velocity, then $d v / d t=-32+k v^2$, where $-32$ feet per second per second is the acceleration due to gravity and $k$ is a constant. Show that the velocity $v$ as a function of time $t$ is $v(t)=-\sqrt{\frac{32}{k}} \tanh (\sqrt{32 k} t)$ by performing the following integration and simplifying the result.

$$\int \frac{d v}{32-k v^2}=-\int d t$$

A car traveling at 45 miles per hour is brought to a stop, at constant deceleration, 132 feet from where the brakes are applied.

Draw the real number line from 0 to 132 , and plot the points found in parts (a) and (b). What can you conclude?

A car traveling at 45 miles per hour is brought to a stop, at constant deceleration, 132 feet from where the brakes are applied.

How far has the car moved when its speed has been reduced to 15 miles per hour?