Methods of Integration Formulas Techniques and Solved Examples
In Mathematics, when we cannot perform general addition operations, we use integration to add values on a large scale.
There are various methods in mathematics to integrate functions.
Integration and differentiation are also a pair of inverse functions similar to addition- subtraction, and multiplication-division.
The process of finding functions whose derivative is given is named anti-differentiation or integration.
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Here’s What Integration is!
Points to Remember:
Types of Integration Maths or the Integration Techniques-
Here’s a list of Integration Methods –
1. Integration by Substitution
2. Integration by Parts
3. Integration by Partial Fraction
4. Integration of Some particular fraction
5. Integration Using Trigonometric Identities
For better understanding here’s what each method is!
1. Integration by Substitution -
We can find the integration by introducing a new independent variable when it is difficult to find the integration of a function.
By changing the independent variable x to t, in a given form of integral function say
(∫f(x))
(∫f(x)), we can transform the integral.
Let’s substitute the value of independent x = g(t) in the integral function ∫f(x),
We get, dx / dt = g’(t)
Or, dx = g’(t) • dt
Thus, from the above substitution ,we get,
I=∫f(x).dx=f(g(t).
g′(t)).dt
I=∫f(x).dx=f(g(t).g′(t)).dt
2. Integration by Parts –
If the integrand function can be represented as a multiple of two or more functions, the integration of any given function can be done by using the Integration by Parts method.
Let us take an integrand function that is equal to f(x)g(x).
In mathematics, Integration by part uses the ILATE rule for selecting the first and second functions in this method.
In mathematics, here’s how integration by parts is represented.
∫f(x).g(x).dx = f(x).∫g(x).dx – ∫(f′(x).∫g(x).dx).dx
Which can be further written as integral of the product of any two functions = (First function × Integral of the second function) – Integral of
(differentiation of the first function)×Integral Of The Second Function
(differentiation of the first function)×Integral Of The Second Function
3. Integration Using Trigonometric Identities –
Trigonometric identities are used to simplify any integral function which consists of trigonometric functions.
It simplifies the integral function so that it can be easily integrated.
There are many trigonometric identities, a few are listed below!
Sin2x = \[\frac {(1-Cos 2x)}{2}\]
Cos2x = \[\frac {(1+ Cos 2x)}{2}\]
4. Integration of Some Particular Function -
Many other standard integrals can be integrated using some important integration formulas.
Here are the six important formulas listed below -
∫ dx/ (x2 – a2) = ½ a log | (x – a) / (x + a) | + c
∫ dx/ (a2 – x2) = ½ a log | (a + x) / (a – x) | + c
∫ dx / (x2 + a2) = 1/a tan–1 (x/a) + c
∫ dx /√ (x2 – a2) = log| x+√(x2 – a2) | + c
∫ dx /√ (a2 – x2) = sin–1 (xa) + c
∫ dx /√ (x2 + a2) = log | x + √(x2 + a2) | + c
Where, c = constant
5. Integration by Partial Fraction -
The partial fraction method is the last method of integration class 12.
In mathematics, rational numbers can be expressed in the form of
p
q
pq
where p and q are integers and where the value of the denominator q is not equal to zero.The ratio of two polynomials is known as a rational fraction and it can be expressed in the form of
p(x)
q(x)
p(x)q(x)
, where the value of p(x) should not be equal to zero.The two forms of partial fraction have been described below-
Proper Partial function
Improper Partial function
What is the proper partial function?
When the degree of the denominator is more than the degree of the numerator, the function is known as a proper partial function.
What is an improper partial function?
When the degree of the denominator is less than the degree of the numerator then the fraction is known as improper partial function. Thus, the fraction can be simplified into parts and can be integrated easily.
Topics Covered in Methods of Integration: Definitions, Types, Examples
Integration is used to add large values in mathematics when the calculations cannot be performed on general operations. There are many methods of integration that are used specifically to solve complex mathematical operations.
The different kinds of methods of integration are: -
Integration by Parts.
Method of Integration Using Partial Fractions.
Integration by Substitution Method.
Integration by Decomposition.
Reverse Chain Rule.
Integration Using Trigonometric Identities.
All methods of integration are important. Integration by parts is one of the best because it is used when a function that has to be integrated is written as a product of two or more. Integration by parts is also known as the product rule of integration and the UV method of integration. When you have to integrate rational functions, a method of integration using partial fractions is used. The reverse chain rule is also one of the easiest and most commonly used methods of integration.
FAQs on Methods of Integration in Calculus
1. What are the main methods of integration?
The main methods of integration are substitution, integration by parts, partial fractions, trigonometric integrals, and trigonometric substitution.
- Substitution (u-substitution) – used when the integrand contains a composite function.
- Integration by parts – used for products of functions.
- Partial fractions – used for rational functions.
- Trigonometric integrals – used for powers of sine and cosine.
- Trigonometric substitution – used for radicals like √(a² − x²).
2. What is the substitution method in integration?
The substitution method (or u-substitution) simplifies an integral by changing variables to make it easier to evaluate.
- Let u = g(x).
- Then du = g'(x) dx.
- Rewrite the integral in terms of u.
Let u = x², so du = 2x dx.
The integral becomes ∫cos(u) du = sin(u) + C = sin(x²) + C.
3. What is integration by parts formula?
The formula for integration by parts is ∫u dv = uv − ∫v du.
- Choose u (usually algebraic or logarithmic).
- Choose dv (usually exponential or trigonometric).
- Differentiate u to get du.
- Integrate dv to get v.
4. When do you use partial fractions in integration?
Use partial fractions when integrating a rational function where the degree of the numerator is less than the denominator.
- Factor the denominator.
- Decompose into simpler fractions.
- Integrate each term separately.
5. What is the formula for integrating powers of x?
The power rule for integration is ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, where n ≠ −1.
- Add 1 to the exponent.
- Divide by the new exponent.
Special case: ∫1/x dx = ln|x| + C.
6. What is trigonometric substitution in integration?
Trigonometric substitution is a method used to evaluate integrals containing radicals like √(a² − x²), √(a² + x²), or √(x² − a²).
- If √(a² − x²), let x = a sinθ.
- If √(a² + x²), let x = a tanθ.
- If √(x² − a²), let x = a secθ.
7. What is the difference between definite and indefinite integrals?
An indefinite integral gives a family of antiderivatives, while a definite integral gives a numerical value representing area.
- Indefinite integral: ∫f(x) dx = F(x) + C.
- Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a).
8. How do you integrate exponential functions?
To integrate exponential functions, use the rule ∫eˣ dx = eˣ + C and ∫aˣ dx = aˣ/ln(a) + C.
- For ∫e^(kx) dx, result is e^(kx)/k + C.
- For ∫a^(kx) dx, result is a^(kx)/(k ln a) + C.
9. What are common mistakes in methods of integration?
Common mistakes in integration techniques include forgetting constants, incorrect substitution, and algebra errors.
- Forgetting the constant of integration + C.
- Not changing dx correctly in substitution.
- Choosing wrong u and dv in integration by parts.
- Not adjusting limits in definite integrals after substitution.
10. Can you give a simple example combining two methods of integration?
Yes, some integrals require combining methods such as substitution and partial fractions.
- Example: ∫x/(x² − 1) dx.
- Let u = x² − 1, so du = 2x dx.
- Rewrite as ½ ∫du/u.
