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Methods of Integration

Methods of Integration include different methods of solving complex and simple problems of integration in calculus. To apply a specific method of integration, first, we need to identify the type of integral involved and then apply the most suitable method of integration to solve it.

In this article, we will explore different methods of integration such as integration by parts, substitution method, method of integration using partial fractions, reverse chain rule among others. We will solve a few examples to understand the applications of these methods of integration.

1. What are Methods of Integration?
2. List of Methods of Integration
3. Integration By Parts
4. Method of Integration Using Partial Fractions
5. Reverse Chain Rule
6. Substitution Method of Integration
7. FAQs on Methods of Integration

What are Methods of Integration?

Before proceeding with the methods of integration, let us first recall the concept of integration. Integration is the process of combining very small strips of a figure to get the area of the whole figure. It gives the area under the curve of a function. We use different methods of integration to find the integral of complex functions. To simplify the integral problems, we need to identify the type of function to be integrated and then apply the integration method which makes it easier to solve. We also use trigonometric formulas and identities as a method of integration to simplify the trigonometric functions under integration.

List of Methods of Integration

We can solve the integrals of functions using the inspection method. But sometimes, this does not work and we need to simplify the function first before evaluating its integral. To simplify these complex functions, we use integration methods in calculus. Given below is the list of the different methods of integration that are useful in simplifying integration problems:

  • Integration by Parts
  • Method of Integration Using Partial Fractions
  • Integration by Substitution Method
  • Integration by Decomposition
  • Reverse Chain Rule
  • Integration Using Trigonometric Identities

Let us explore some of these methods of integration and see their formulas which will help in identifying the correct integration method to apply.

Integration By Parts

Integration by parts is one of the important methods of integration. It is used when the function to be integrated is written as a product of two or more functions. It is also called the product rule of integration and uv method of integration. If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:

∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C

Here f(x) is the first function and g(x) is the second function.

For integrating by parts, the choice of the first function is done on the basis of the sequence given below. This method is also commonly known as the ILATE or LIATE method of integration which is abbreviated of:

  • I – Inverse Trigonometric Function
  • L – Logarithmic Function
  • A – Algebraic Function
  • T – Trigonometric Function
  • E – Exponential Function

Method of Integration Using Partial Fractions

This method of integration is used to integrate rational functions. It is used to decompose the denominator of the rational function and convert it into simpler multiple rational functions. Integration by partial fractions is one of important methods of integration.The formula to integrate rational functions of the form f(x)/g(x) is:

∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx

where

  • f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
  • g(x) = q(x).s(x)

Now, different forms of rational functions are decomposed using particular forms of partial fractions to make the calculation easy and simple. Refer to our page on integration by partial fractions to know about each form and how to simplify the functions.

Reverse Chain Rule

The reverse chain rule is one of the easiest and popular methods of integration as it is the reverse process of the chain rule in differentiation. Here, we identify the derivative in the function to be integrated. This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,

∫g'(f(x)) f'(x) dx = g(f(x)) + C

Integration by Substitution Method of Integration

The substitution method is also commonly called the u-substitution method of integration. This method enables us to change the variable of integration to simplify the function. It is similar to the reverse chain rule. For instance, we have integration of the form ∫g(f(x)) dx. Then we can substitute the f(x) with another variable by assuming f(x) = u. We differentiate f(x) = u which implies f'(x) dx = du ⇒ dx = du/h(u), where h(u) = f'(x) using f(x) = u-substitution. Please note that if we change the variable of integration, then it has to be changed throughout the integral. Hence the formula for integration using the substitution method becomes:

∫g(f(x)) dx = ∫g(u)/h(u) du

Important Notes on Methods of Integration

  • The functions to be integrated can be decomposed into sum or difference of functions, whose individual integrals are known.
  • Add the constant of integration always after determining the integral of the function.

Related Topics on Methods of Integration

Methods of Integration Examples

  1. Example 1: Evaluate the integral ∫ [1/(x2 + 5x + 6)] dx using one of the methods of integration.

    Solution: To determine the value of ∫ [1/(x2 + 5x + 6)] dx, we will factorize (x2 + 5x + 6)

    x2 + 5x + 6 = x2 + 3x + 2x + 6

    = x(x + 3) + 2(x + 3)

    = (x + 2)(x + 3)

    To simplify 1/(x2 + 5x + 6) = 1/[(x + 2)(x + 3)], we will assume

    1/[(x + 2)(x + 3)] = A/(x + 2) + B/(x + 3)

    = (Ax + 3A + Bx + 2B)/[(x + 2)(x + 3)]

    Comparing the coefficients we have,

    3A + 2B = 1 and A + B = 0. Solving these two equations, we have A = 1, B = -1

    Hence, ∫ [1/(x2 + 5x + 6)] dx = ∫ 1/[(x + 2)(x + 3)] dx

    = ∫1/(x + 2) dx – ∫1/(x + 3) dx

    = ln |x + 2| – ln |x + 3| + C

    Answer: ∫ [1/(x2 + 5x + 6)] dx = ln |x + 2| – ln |x + 3| + C

  2. Example 2: Determine the value of ∫x ln x dx using the by parts method of integration.

    Solution: To solve ∫x ln x dx using by parts method of integration, we will consider the sequence in ILATE and assume ln x as the first function (because it is a logarithmic function) and x as the second function (it is an algebraic function).

    ∫x ln x dx = ln x ∫x dx – ∫[(ln x)’ ∫x dx] dx

    = (x2/2) ln x – ∫(1/x)(x2/2) dx

    = (x2/2) ln x – ∫(x/2) dx

    = (x2/2) ln x – x2/4 + C

    Answer: Hence ∫x ln x dx = (x2/2) ln x – x2/4 + C

Practices Questions on Methods of Integration

FAQs on Methods of Integration

What are the Methods of Integration in Calculus?

Given below is the list of the different methods of integration that are useful in simplifying integration problems:

  • Integration by Parts
  • Method of Integration Using Partial Fractions
  • Integration by Substitution Method
  • Integration by Decomposition
  • Reverse Chain Rule
  • Integration Using Trigonometric Identities

How many Types of Methods of Integration are there?

There are many methods of integration that we use but the most common ones are 5, namely Integration by Parts, Method of Integration Using Partial Fractions, Integration by Substitution Method, Integration by Decomposition, and Reverse Chain Rule.

Which Method of Integration is the Counterpart of the Chain Rule of Differentiation?

Reverse Chain Rule Method of Integration is the counterpart of the Chai Rule of Differentiation and it is the exact reverse of the chain rule method. This method of integration can be used if a function and its derivative are a part of the entire function to be integrated.

What are the Most Commonly Used Integration Methods?

The most commonly used Integration methods are Integration by Parts, Method of Integration Using Partial Fractions, u-substitution method, Integration by Decomposition, and Reverse Chain Rule.

When Do We Use Methods of Integration?

We use methods of integration when the function to be integrated is complex and hence simplify the function into simple forms whose integration is known.

Are Methods of Integration Same as the Methods of Anti-differentiation?

Yes the methods of integration and the methods of differentiation are the same as integration is nothing but the reverse process of differentiation.

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