# Basic Integration Formulas And Examples Pdf

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In mathematics , an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data.

Integration is the reverse process of differentiation, so the table of basic integrals follows from the table of derivatives. Below is a list of top integrals. Calculation of integrals using the linear properties of indefinite integrals and the table of basic integrals is called direct integration. All three integrals can be evaluated using the integration table.

## Differentiation & Integration Formulas With Examples PDF

In this section, we use some basic integration formulas studied previously to solve some key applied problems. It is important to note that these formulas are presented in terms of indefinite integrals. Although definite and indefinite integrals are closely related, there are some key differences to keep in mind. A definite integral is either a number when the limits of integration are constants or a single function when one or both of the limits of integration are variables.

An indefinite integral represents a family of functions, all of which differ by a constant. As you become more familiar with integration, you will get a feel for when to use definite integrals and when to use indefinite integrals. You will naturally select the correct approach for a given problem without thinking too much about it. However, until these concepts are cemented in your mind, think carefully about whether you need a definite integral or an indefinite integral and make sure you are using the proper notation based on your choice.

Recall the integration formulas given in the table in Antiderivatives and the rule on properties of definite integrals. The first step is to rewrite the function and simplify it so we can apply the power rule:. The net change theorem considers the integral of a rate of change. It says that when a quantity changes, the new value equals the initial value plus the integral of the rate of change of that quantity.

The formula can be expressed in two ways. The second is more familiar; it is simply the definite integral. The new value of a changing quantity equals the initial value plus the integral of the rate of change:. Subtracting F a F a from both sides of the first equation yields the second equation. Since they are equivalent formulas, which one we use depends on the application.

The significance of the net change theorem lies in the results. Net change can be applied to area, distance, and volume, to name only a few applications. Net change accounts for negative quantities automatically without having to write more than one integral. We looked at a simple example of this in The Definite Integral.

Suppose a car is moving due north the positive direction at 40 mph between 2 p. We can graph this motion as shown in Figure 5. Just as we did before, we can use definite integrals to calculate the net displacement as well as the total distance traveled.

The net displacement is given by. Thus, at 5 p. The total distance traveled is given by. Therefore, between 2 p. To summarize, net displacement may include both positive and negative values. In other words, the velocity function accounts for both forward distance and backward distance. To find net displacement, integrate the velocity function over the interval.

Total distance traveled, on the other hand, is always positive. To find the total distance traveled by an object, regardless of direction, we need to integrate the absolute value of the velocity function.

Use Example 5. The total distance traveled includes both the positive and the negative values. Therefore, we must integrate the absolute value of the velocity function to find the total distance traveled. To continue with the example, use two integrals to find the total distance. First, find the t -intercept of the function, since that is where the division of the interval occurs. Set the equation equal to zero and solve for t.

To find the total distance traveled, integrate the absolute value of the function. Thus, we have. So, the total distance traveled is 41 6 41 6 m.

The net change theorem can be applied to the flow and consumption of fluids, as shown in Example 5. Express the problem as a definite integral, integrate, and evaluate using the Fundamental Theorem of Calculus. The limits of integration are the endpoints of the interval [ 0 , 1 ]. We have. As we saw at the beginning of the chapter, top iceboat racers Figure 5. Andrew is an intermediate iceboater, though, so he attains speeds equal to only twice the wind speed.

Suppose Andrew takes his iceboat out one morning when a light 5-mph breeze has been blowing all morning. As Andrew gets his iceboat set up, though, the wind begins to pick up. In other words, the wind speed is given by. To figure out how far Andrew has traveled, we need to integrate his velocity, which is twice the wind speed. Substituting the expressions we were given for v t , v t , we get.

Under these conditions, how far from his starting point is Andrew after 1 hour? The graphs of even functions are symmetric about the y -axis. The symmetry appears in the graphs in Figure 5.

Graph a shows the region below the curve and above the x -axis. We have to zoom in to this graph by a huge amount to see the region. Graph b shows the region above the curve and below the x -axis. The signed area of this region is negative. Both views illustrate the symmetry about the y -axis of an even function.

To verify the integration formula for even functions, we can calculate the integral from 0 to 2 and double it, then check to make sure we get the same answer.

The graph is shown in Figure 5. Use basic integration formulas to compute the following antiderivatives or definite integrals. Write an integral that expresses the increase in the perimeter P s P s of a square when its side length s increases from 2 units to 4 units and evaluate the integral. A regular N -gon an N -sided polygon with sides that have equal length s , such as a pentagon or hexagon has perimeter Ns. Write an integral that expresses the increase in perimeter of a regular N -gon when the length of each side increases from 1 unit to 2 units and evaluate the integral.

The Pentagon in Washington, DC, has inner sides of length ft and outer sides of length ft. Write an integral to express the area of the roof of the Pentagon according to these dimensions and evaluate this area.

A dodecahedron is a Platonic solid with a surface that consists of 12 pentagons, each of equal area. By how much does the surface area of a dodecahedron increase as the side length of each pentagon doubles from 1 unit to 2 units?

An icosahedron is a Platonic solid with a surface that consists of 20 equilateral triangles. By how much does the surface area of an icosahedron increase as the side length of each triangle doubles from a unit to 2 a units? Write an integral that quantifies the change in the area of the surface of a cube when its side length doubles from s unit to 2 s units and evaluate the integral.

Write an integral that quantifies the increase in the volume of a cube when the side length doubles from s unit to 2 s units and evaluate the integral. Write an integral that quantifies the increase in the surface area of a sphere as its radius doubles from R unit to 2 R units and evaluate the integral.

Write an integral that quantifies the increase in the volume of a sphere as its radius doubles from R unit to 2 R units and evaluate the integral. A ball is thrown upward from a height of 1.

Neglecting air resistance, solve for the velocity v t v t and the height h t h t of the ball t seconds after it is thrown and before it returns to the ground. The area A t A t of a circular shape is growing at a constant rate.

A spherical balloon is being inflated at a constant rate. Find the change in height between 5 min and 10 min. The following table lists the electrical power in gigawatts—the rate at which energy is consumed—used in a certain city for different hours of the day, in a typical hour period, with hour 1 corresponding to midnight to 1 a. Find the total amount of energy in gigawatt-hours gW-h consumed by the city in a typical hour period.

The average residential electrical power use in hundreds of watts per hour is given in the following table. The data in the following table are used to estimate the average power output produced by Peter Sagan for each of the last 18 sec of Stage 1 of the Tour de France. The data in the following table are used to estimate the average power output produced by Peter Sagan for each min interval of Stage 1 of the Tour de France.

If Earth has mass 5. On wet asphalt, it is approximately 2. Given that 1 mph corresponds to 0. Find the corresponding distances if the surface is slippery wet asphalt.

John is a year old man who weighs lb. If an oatmeal cookie has 55 cal and John eats 4 t cookies during the t th hour, how many net calories has he lost after 3 hours riding his bike? Sandra is a year old woman who weighs lb. Her caloric intake from drinking Gatorade is t calories during the t th hour. What is her net decrease in calories after walking for 3 hours? A motor vehicle has a maximum efficiency of 33 mpg at a cruising speed of 40 mph.

The efficiency drops at a rate of 0. What is the efficiency in miles per gallon if the car is cruising at 50 mph?

## 4.6: Integration Formulas and the Net Change Theorem

The fundamental use of integration is as a continuous version of summing. But, paradoxically, often integrals are computed by viewing integration as essentially an inverse operation to differentiation. That fact is the so-called Fundamental Theorem of Calculus. Home Threads Index About. Basic integration formulas. Thread navigation Calculus Refresher Previous: Graphing rational functions, asymptotes Next: The simplest integration substitutions Similar pages The simplest integration substitutions Integration substitutions Area and definite integrals Length of curves Numerical integration Averages and weighted averages Centers of mass centroids Volumes by cross sections Volume of surfaces of revolution Integration by parts More similar pages.

Integration and Differentiation are two fundamental concepts in calculus, which studies the change. Calculus has a wide variety of applications in many fields such as science, economy or finance, engineering and etc. Differentiation is the algebraic procedure of calculating the derivatives. Derivative of a function is the slope or the gradient of the curve graph at any given point. Gradient of a curve at any given point is the gradient of the tangent drawn to that curve at the given point. Integration is the process of calculating either definite integral or indefinite integral. When a specific interval is not given, it is known as indefinite integral.

Basic formulas. Most of the following basic formulas directly follow the differentiation rules. Example 1: Evaluate. Using formula 4 from the preceding list, you find that. Example 2: Evaluate. Because using formula 4 from the preceding list yields. Example 3: Evaluate.

## Integration Formula Sheet - Chapter 7 Class 12 Formulas

Over Integrals Served. Right click on any integral to view in mathml. The integral table in the frame above was produced TeX4ht for MathJax using the command sh.

In this section, we use some basic integration formulas studied previously to solve some key applied problems. It is important to note that these formulas are presented in terms of indefinite integrals. Although definite and indefinite integrals are closely related, there are some key differences to keep in mind.

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### Integration Formula Sheet - Chapter 7 Class 12 Formulas

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du = 3 dx. du = dx. / cos(x) sin(2x) + sin(x) cos(2x) dx = / sin (x + 2x) dx. = / sin (3x) dx. = / sin (u) du.

#### Download PDF of Differentiation & Integration Formulas With Examples from cengage.com

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10.12.2020 at 21:52

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In this section, we use some basic integration formulas studied previously to solve some key applied problems.