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You must be aware of the formulas for calculating the areas of various geometrical objects, such as a square, rectangle, triangle, and so on.

But how can you calculate the areas under numerous complicated curves using only the fundamental areas that are already known to you?

Most likely, you'll consider splitting the provided area into some basic forms that you are already familiar with, and then adding up the areas of the various shapes to approximate the final result.

In mathematics, this process is referred to as integration, and it is studied under the branch of integral calculus. Online integration calculator with steps has many benefits and it is very useful for learning and practice.

To learn more about integration, its significance and various integration techniques read the article below.

Related: Everything you need to know about integration by partial fraction.

Integration

Integral calculus is the branch of calculus that is concerned with the area bounded by a curve and the x axis of the coordinate system.

The symbol ‚®úrepresents the concept of integration. As the slope of an equation is obtained by a differentiation method.

However, in contrast when you integrate, you obtain the area that exists between the equation and the x-axis of the graph.

It is possible to define the integral or anti-derivative of a function as another function whose derivative is equal to the original function; this is known as the anti-derivative of the function.

Related: Also find other Integration tools on this website like polar double integral calculator and triple integral calculator online for practice.

Types of Integration

The integration is of two distinct types,

  1. Indefinite Integrals
  2. Definite Integrals

The indefinite integrals comprise the calculation under defined boundaries or limits within which the function is calculated.

In contrast, an indefinite calculus doesn't have defined limits i.e. no upper and lower bounds and always contains a constant value.

You can find definite integral calculator step by step and indefinite integral calculator step by step on this website.

Significance of Integration

The definite as well as indefinite integrals have implications in real life and that's why they are really important concepts.

The definite integrals, for instance, are used to calculate the length of the curve.

The longitude of the arc is initially estimated using line segments, which create a sum of Riemann.

To obtain the surface area of a revolution surface, the notions used for determining the arc length may be generalized.

It is typically difficult to assess the integrals given by both the arc length and the surface area formula.

Same as the definite integrals are used to obtain a surface under a curve, the area between two curves can also be found in the same way by definite integrals.

Integrate function differences to determine the area between two function-defined curves.

Related: Also find other useful online calculators like shell method volume calculator and washer method integral calculator on this website.

Moreover, in engineering and physics, a number of practical uses of the integral are frequent.

Depending on the density function, certain integrals may be used to estimate the mass of an item. The work can also be computed via integration of a force function or, as in pumping problems, the force of gravity.

The integrals can also be utilized in calculating the force of an object immersed in a liquid.

Several other fields and calculations where the integral calculus find its implications and is really important are includes:

  • The area between two curves
  • Centre of mass
  • Kinetic energy
  • Surface area
  • Work
  • Distance, velocity and acceleration
  • The average value of a function
  • Volume
  • Probability

Related: Also find disk method cross sectional calculator to calculate cross sectional integral functions.

Methods of Integration

An individual integration problem can be approached using a variety of techniques, strategies and different integration methods.

It is possible to remember a number of well-known integration formula. All of these, on the other hand, will not be applicable in all circumstances.

Now the basic 2 techniques of integration will be discussed. However, let’s start with substitution, which is a method that we utilized in differential calculus as well.

Integration by Substitution

Substitution method, according to some mathematicians, is the polar opposite of the chain rule, which is employed in differentiation.

By employing this approach, it is possible to convert an integrand that appears complex to a known simple form the integration formula of which is already identified.

If you put in enough effort and have a thorough grasp of the integration formulas, you will be able to determine the substitutions in integration to make.

The substitution method is used whenever we have a function with its derivative under the integral for calculation.

It is also known as the U-substitution method as we substitute variables with “u” most of the time. The rule for substitution integral is given as follow:

$$\int f(u(x))u^′(x)dx=\int f(u)du$$

where

$$u=u(x)$$

here, integral on the left hand side is expressed in variable “x” while the integral on the right hand side is substituted “x” variable by “u”

For example:

Find the integral :

$$\int(3x+2)5dx$$

Solution:

We will substitute

$$u=3x+2$$

then

$$du=d(3x+2)=3dx$$

So the differential dx is given by

$$dx=\frac{du}{3}$$

Plug all this in the integral:

$$\int(3x+2)^5dx = \int u^5 \frac{du}{3}= \frac{1}{3} \int u^5du$$

$$\frac{1}{3}⋅\frac{u^6}{6}+C$$

$$\frac{u^6}{18}+C$$

$$\frac{(3x+2)^6}{18} + C$$

Integration by Parts

Integration by parts is a unique rule that may be used for the integration of two distinct functions products.

More specifically, this is a particular integration approach that is employed to multiply two functions together.

This approach is not only useful for the product of functions, but it can also be utilized for a variety of other calculations as well.

The indefinite or definite integration is accomplished using this specific approach by extending the differential of a product function and expressing the original terms alone in terms of integrals.

We can use integration by parts in complex integral functions where two or more different functions are multiplied. The key formula for integration by parts are given as follow:

$$\frac udv = uv− \frac vdu$$

here,

u = function of u(x), v = function of v(x)

dv = variable of dv, du = variable of du

The main point for integration by parts is to selecting u and dv correctly in following way:

I = Inverse Functions

L = Logarithmic Functions

A = Algebraic Functions

T = Trigonometric Function

E = Exponential Function

The closer function should be selected as a u(x) in the following form of ILATE.

For example:

Resolve the integral

$$\int x cos2xdx$$

Solution:

we choose

$$u = x$$

$$dv = cos2xdx$$

So,

$$du=dv$$

$$v = \int cos2xdx = \frac{1}{2}.sin2x$$

Now we put following in our formula of integration by parts

$$\int udv=uv−\int vdu$$

$$\int xcos2x dx=x⋅\frac{1}{2} sin2x − \int\frac{1}{2}sin2xdx$$

$$\frac{x}{2}sin2x−\frac{1}{2}\int sin2xdx $$

$$\frac{x}{2}sin2x−\frac{1}{2}⋅(−\frac{1}{2}cos2x)+C$$

$$\frac{x}{2}sin2x+\frac{1}{4}cos2x + C$$

Integration along with differentiation are perhaps the most important topics in calculus and both has their own significance. These concepts are widely used in our routine life and it is very important for students to learn these. We hope you liked this article, do share with your friends on social media. Cheers!

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