Methods of Integration

Integration can be defined as the summation of values when the number of terms tends to infinity. It is used to unite a part of the whole. Integration is just the reverse of differentiation and has various applications in all spheres, such as physics, chemistry, space, engineering, etc.

Integration of a function or a curve can be used to find useful information, such as the area under the curve or volume of the curve, etc. Integration may be of 2 types, which are definite and indefinite integration, depending upon whether the limits of integration are mentioned or not.

We know that integration is represented using the symbol ∫ over a function f(x) as follows:

∫f(x)dx 

Integration By Substitution

The substitution method is used when we find it difficult to integrate a function as it is. In this method, a certain term in the function is substituted as a new variable, and the whole function is changed to a new function of a new variable.

Example: Solve ∫(x - 4) dx.

Solution: 

Given 

Substitute (x-4) = u

Differentiate both sides w.r.t x to get 

dx = d

Thus 

Again putting u = (x-4)

Integration using Trigonometric Identities

Integration using Trigonometric identites involves integrating the given function by transforming it using trigonometric identities. The value of the given function is substituted using some other function that is derived by using trigonometric identities.

Example: Solve 

Solution:

Given 

We know that cos(2x) = 2cos²x -1

2cos²x = cos(2x)+1

Substituting the value of 2cos²x, we get 

Integration by Parts

This method is used in cases where the function to be integrated is a product of two or more functions. Let f(x) = g(x)h(x), then f(x) can be integrated by parts by using the below formula:

∫g(x) · h(x) · dx = g(x) · ∫h(x) dx – ∫(g′(x) · ∫h(x) dx) dx

The sequence of h(x) and g(x) should be decided using the ILATE rule which tells the priority of functions and stands for Inverse trigonometric, logarithmic, algebraic, trigonometric, and exponential functions. This means that inverse trigonometric function should be written before logarithmic, logarithmic should be written before algebraic and so on.

Example:Solve

Solution:

Given 

From ILATE rule, we know that logarithmic function has higher priority than algebraic function. Hence,

Using integration by parts,

Integration by Partial Fraction

If the function to be integrated is of the form f(x) = g(x)/h(x) where g(x) and h(x) are polynomials, then we use the method of partial fraction. There are multiple cases in partial fractions depending upon the type of f(x).

In all these cases, we need to take the LCM of the partial fractions to make the denominator the same. After that, we compare the numerator on the LHS and RHS. Then substitute the suitable value of x in order to make any one part of the numerator zero and determine the value of A, B and C.

Example: Integrate the function f(x) = x/(x-2)(x+3).

Solution:

Given f(x) = x/(x-2)(x+3)

It is of the form 

So it can be written as 

Taking LCM on RHS, we get,

Comparing numerators on LHS and RHS:

Put x = 2 to get,

2 = 5A or A = 2/5

similarly put x = -3 to get

-3 = -5B or B = 3/5

Thus 

Integration of Some Special Functions

In Mathematics, we have some special functions which have pre-defined integration formulas. These functions and their integration are shown below:

Solved Examples

Example 1: Solve .

Solution:

Given: I = \int \log x~dx = \int \log x.1~dx 

The functions are already written according to ILATE rule. Using integration by parts we get,

Example 2: Solve 

Solution:

Given: I = \int x \sin x~dx

The functions are already written according to ILATE rule. Using integration by parts we get,

Example 3: Solve 

Solution:

Given: 

Substitute (2x³+1) = u 

Differentiate both sides w.r.t x to get 

6x² dx = du

x²dx = du/6

Rewriting the given function as 

Example 4: Solve 

Solution:

Given 

We know that cos(2x) = 1 - 2sin²x

(1-cos2x)/2 = sin²x

Substituting the value of sin²x, we get 

Example 5: Solve .

Solution:

Given:

Using 

Practice Problems

Problem 1: Calculate the following integrals:

• ∫(3x² + 2x - 5) dx
• ∫(e^x + 1) dx
• ∫(sin(x) + cos(x)) dx

Problem 2: Use the substitution method to evaluate the following integrals:

• ∫(2x + 1)³ dx
• ∫(x² + 4x + 3)√(x³ + 6x² + 9x) dx
• ∫2x cos(x²) dx

Problem 3: Apply integration by parts to solve the following integrals:

• ∫x ln(x) dx
• ∫x² sin(x) dx
• ∫e^x cos(x) dx

Problem 4: Use partial fraction decomposition to integrate:

• ∫(3x² + 2)/(x³ + 4x² + 4x) dx
• ∫(x³ - 2x² + 5x - 1)/(x² - 3x + 2) dx
• ∫(4x³ - 2x² + 3)/(x⁴ + 2x² + 1) dx