Different Integration Formulas
Various types of integration methods are used to solve different types of integral questions. Each method is a standard result and can be considered a formula. Some of the important methods are discussed below in this article. Let’s check the three important integration methods.
- Integration by Parts Formula
- Integration by Substitution Formula
- Integration by Partial Fractions Formula
Integration by Parts Formula
Integration by Parts Formula is applied when the given function is easily described as the product of two functions. The integration by Parts formula used in mathematics is given below,
∫ f(x) g(x) dx = f(x) ∫g(x) dx – ∫ (∫f'(x) g(x) dx) dx + C
Example: Calculate ∫ xex dx
Solution:
∫ xex dx is of the form ∫ f(x) g(x) dx
let f(x) = x and g(x) = ex
we know that, ∫ f(x) g(x) dx = f(x) ∫g(x) dx – ∫ (∫f'(x) g(x) dx) dx + C
∫ xex dx = x ∫ex dx – ∫( 1 ∫ex dx) dx+ c
= xex – ex + c
Integration by Substitution Formula
Integration by Substitution Formula is applied when a function is a function of another function. i.e. let I = ∫ f(x) dx, where x = g(t) such that dx/dt = g'(t), then dx = g'(t)dt
Now, I = ∫ f(x) dx = ∫ f(g(t)) g'(t) dt
Example: Evaluate ∫ (4x +3)3 dx
Solution:
Let u = (4x+3) ⇒ du = 4 dx
∫ (4x +3)3 dx
= 1/4 ∫(u)3 du
= 1/4. u4 /5
= u4 /20
= 4x +3)4/20
Integration by Partial Fractions Formula
Integration by Partial Fractions Formula is used when the integral of P(x)/Q(x) is required and P(x)/Q(x) is an improper fraction, such that the degree of P(x) is less than the (<) the degree of Q(x), then the fraction P(x)/Q(x) is written as
P(x)/Q(x) = R(x) + P1(x)/ Q(x)
where
- R(x) is a polynomial in x
- P1(x)/ Q(x) is a proper rational function
Now the integration of R(x) + P1(x)/ Q(x) is easily calculated using the formulas discussed above.
Integration Formulas
Integration Formulas are the basic formulas that are used to solve various integral problems. They are used to find the integration of algebraic expressions, trigonometric ratios, inverse trigonometric functions, and logarithmic and exponential functions. These integration formulas are very useful for finding the integration of various functions.
Integration is the inverse process of differentiation, i.e. if d/dx (y) = z, then ∫zdx = y. Integration of any curve gives the area under the curve. We find the integration by two methods Indefinite Integration and Definite Integration. In indefinite integration, there is no limit to the integration whereas in definite integration there is a limit under which the function is integrated.
Let us learn about these integral formulas, and their classification, in detail in this article.
Table of Content
- Integral Calculus
- What are Integration Formulas?
- Integration Formulas of Trigonometric Functions
- Integration Formulas of Inverse Trigonometric Functions
- Advanced Integration Formulas
- Different Integration Formulas
- Application of Integrals
- Definite Integration Formula
- Indefinite Integration Formula