In this given exercise, use the Second Fundamental Theorem of Calculus to find $F^{\prime}(x)$.

$$F(x)=\int_1^x \frac{t^2}{t^2+1} d t$$

The demand equation for a product is

$$p=\frac{90,000}{400+3 x} \text {. }$$

Find the average price $p$ on the interval $40 \leq x \leq 50$.

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.

$$\int_{-1}^1 e^{-x^2} d x$$

In this given exercise, use the Second Fundamental Theorem of Calculus to find $F^{\prime}(x)$.

$$F(x)=\int_{-2}^x\left(t^2-2 t\right) d t$$

In this exercise, evaluate the definite integral. Use a graphing utility to verify your result.

$$\int_{-1}^1 x\left(x^2+1\right)^3 d x$$

The table shows the velocities (in miles per hour) of two cars on an entrance ramp to an interstate highway. The time $t$ is in seconds.

Approximate the distance traveled by each car during the 30 seconds. Explain the difference in the distances.

The table shows the velocities (in miles per hour) of two cars on an entrance ramp to an interstate highway. The time $t$ is in seconds.

Use the regression capabilities of a graphing utility to find quadratic models for the data in part (a).

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

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

The table shows the velocities (in miles per hour) of two cars on an entrance ramp to an interstate highway. The time $t$ is in seconds.

Rewrite the table, converting miles per hour to feet per second.

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

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

Find the time required for an object to cool from $300^{\circ} \mathrm{F}$ to $250^{\circ} \mathrm{F}$ by evaluating

$$t=\frac{10}{\ln 2} \int_{250}^{300} \frac{1}{T-100} d T$$

where $t$ is time in minutes.

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

$$\int t\sqrt[3]{t-4}dt$$

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).

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