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Indefinite Integrals

Calculus 1 / Ab

116 Practice Problems

06:11
Calculus of a Single Variable

Solving a Differential Equation In Exercises $7-10$ , find the general solution of the differential
equation and check the result by differentiation.
$\frac{d y}{d t}=9 t^{2}$

Integration
Antiderivatives and Indefinite Integration
02:24
Calculus of a Single Variable

Think About It Find the general solution of $f^{\prime}(x)=-2 x \sin x^{2}$

Integration
Antiderivatives and Indefinite Integration
Allan H.
02:56
Calculus of a Single Variable

True or False? In Exercises $73-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
$\int f(x) g(x) d x=\left(\int f(x) d x\right)\left(\int g(x) d x\right)$

Integration
Antiderivatives and Indefinite Integration
Allan H.
02:04
Calculus of a Single Variable

True or False? In Exercises $73-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $F(x)$ and $G(x)$ are antiderivatives of $f(x),$ then

$$F(x)=G(x)+C$$

Integration
Antiderivatives and Indefinite Integration
Allan H.
03:05
Calculus of a Single Variable

True or False? In Exercises $73-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
Each antiderivative of an $n$ th-degree polynomial function is an $(n+1)$ th-degree polynomial function.

Integration
Antiderivatives and Indefinite Integration
Allan H.
03:39
Calculus of a Single Variable

Acceleration Assume that a fully loaded plane starting from rest has a constant acceleration while moving down a runway. The plane requires 0.7 mile of runway and a speed of 160 miles per hour in order to lift off. What is the plane's acceleration?

Integration
Antiderivatives and Indefinite Integration
Allan H.
22:28
Calculus of a Single Variable

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

(a) How far has the car moved when its speed has been reduced to 30 miles per hour?
(b) How far has the car moved when its speed has been reduced to 15 miles per hour?
(c) Draw the real number line from 0 to $132 .$ Plot the points found in parts (a) and (b). What can you conclude?

Integration
Antiderivatives and Indefinite Integration
Allan H.
05:41
Calculus of a Single Variable

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

(a) Find the velocity and position functions for the particle.
(b) Find the values of $t$ for which the particle is at rest.

Integration
Antiderivatives and Indefinite Integration
Allan H.
09:15
Calculus of a Single Variable

Repeat Exercise 65 for the position function
$x(t)=(t-1)(t-3)^{2}, 0 \leq t \leq 5$

Integration
Antiderivatives and Indefinite Integration
Allan H.
01:50
Calculus of a Single Variable

Escape Velocity The minimum velocity required for an object to escape Earth's gravitational pull is obtained from the solution of the equation

$\int v d v=-G M \int \frac{1}{y^{2}} d y$

where $v$ is the velocity of the object projected from Earth, $y$ is the distance from the center of Earth, $G$ is the gravitational constant, and $M$ is the mass of Earth. Show that $v$ and $y$ are related by the equation

$v^{2}=v_{0}^{2}+2 G M\left(\frac{1}{y}-\frac{1}{R}\right)$

where $v_{0}$ is the initial velocity of the object and $R$ is the radius of Earth.

Integration
Antiderivatives and Indefinite Integration
Allan H.
02:12
Calculus of a Single Variable

grand canyon The Grand Canyon is 1800 meters deep at its deepest point. A rock is dropped from
the rim above this point. How long will it take the rock to hit the canyon floor?

Integration
Antiderivatives and Indefinite Integration
Allan H.
03:49
Calculus of a Single Variable

Vertical Motion In Exercises $60-62$ , assume the accleration of the object is $a(t)=-9.8$ meters per second per second. (Neglect air resistance.)

A baseball is thrown upward from a height of 2 meters with an initial velocity of 10 meters per second. Determine its maximum height.

Integration
Antiderivatives and Indefinite Integration
Allan H.
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