Algebraic expressions, trigonometric ratios, inverse trigonometric functions, logarithmic and exponential functions can all be integrated using integration formulas. The basic functions for which the derivatives were produced are obtained by integrating functions. These integration formulas are used to obtain a function’s ant derivative. We acquire a family of functions in I when we differentiate a function f in an interval I. We can determine the function f if we know the values of functions in I. Integration is the opposing process of differentiation. Let’s take it a step further and look at the integration formulas that are employed in integration procedures.
Integration Formula- What is it?
The following sets of formulas are a general presentation of the integration formulas. Basic integration formulas, integration of trigonometric ratios, inverse trigonometric functions, the product of functions, and a more complex set of integration formulas are all included. Integration is essentially a method of joining the parts to create the whole. It is differentiation’s opposite action. The fundamental integration equation is thus f'(x) dx = f(x) + C. The following integration formulas are generated using this.
Integration Formulas PDF Class 12
The calculation of an integral is known as integration. Integrals are used in arithmetic to calculate a variety of useful quantities such as areas, volumes, displacement, and so on. When we talk about integrals, we usually mean definite integrals. For ant derivatives, indefinite integrals are utilised. Apart from differentiation, integration is one of the two major calculus subjects in mathematics that measures the rate of change of any function with regard to its variables. It’s a broad topic that’s covered in upper-level classes like Class 11 and 12.
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Integration Formulas PDF
Integration All Formulas PDF
Download All Basic INTEGRATION-FORMULAE
Integration Formulas for Class 12 Students
The most common integration formulas include:
- Constant rule: ∫c dx = cx + C, where c is a constant.
- Power rule: ∫x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1.
- Sum and difference rules: ∫(f(x) ± g(x)) dx = ∫f(x) dx ± ∫g(x) dx.
- Constant multiple rule: ∫c * f(x) dx = c * ∫f(x) dx.
- Trigonometric functions: ∫sin(x) dx = -cos(x) + C, ∫cos(x) dx = sin(x) + C, ∫sec^2(x) dx = tan(x) + C, ∫sec(x)tan(x) dx = sec(x) + C.
- Inverse trigonometric functions: ∫csc^2(x) dx = -csc(x)cot(x) + C, ∫sec(x)cot(x) dx = ln|sec(x) + tan(x)| + C.
- Logarithmic functions: ∫(1/x) dx = ln|x| + C, ∫ln(x) dx = xln(x) – x + C.
- Exponential functions: ∫e^x dx = e^x + C.
- Integration by substitution: ∫f(g(x))g'(x) dx = ∫f(u) du, where u = g(x).
- Integration by parts: ∫u dv = uv – ∫v du, where u and v are functions of x.
- Trigonometric substitution: ∫(a^2 – x^2)^(1/2) dx = a ∫sec^2(θ) dθ, where x = a sin(θ) or x = a tan(θ)
- partial fraction decomposition: ∫(f(x)/(ax+b)) dx = ln|ax+b| + C
Note: C is the constant of integration.
Integration Formulas List
The basic integral formulas are given below:
- ∫ 1 dx = x + C
- ∫ a dx = ax+ C
- ∫ xn dx = ((xn+1)/(n+1))+C ; n≠1
- ∫ sin x dx = – cos x + C
- ∫ cos x dx = sin x + C
- ∫ sec2x dx = tan x + C
- ∫ cosec2x dx = -cot x + C
- ∫ sec x (tan x) dx = sec x + C
- ∫ cosec x ( cot x) dx = – cosec x + C
- ∫ (1/x) dx = ln |x| + C
- ∫ ex dx = ex+ C
- ∫ ax dx = (ax/ln a) + C ; a>0, a≠1
- ∫ tanx.dx =log|secx| + C
- ∫ cotx.dx = log|sinx| + C
- ∫ secx.dx = log|secx + tanx| + C
- ∫ cosecx.dx = log|cosecx – cotx| + C
Integration Formulas of Inverse Trigonometric Functions with limits
- ∫ 1/(1 +x2 ).dx = -cot-1x + C
- ∫ 1/x√(x2 – 1).dx = sec-1x + C
- ∫ 1/x√(x2 – 1).dx = -cosec-1 x + C
- ∫1/√(1 – x2).dx = sin-1x + C
- ∫ /1(1 – x2).dx = -cos-1x + C
- ∫1/(1 + x2).dx = tan-1x + C
List of Difficult Integration Formulas
- ∫ √(x2 + a2 ).dx =1/2.x.√(x2 + a2 )+ a2/2 . log|x + √(x2 + a2 )| + C
- ∫1/(x2 + a2).dx = 1/a.tan-1x/a + C
- ∫1/√(x2 – a2)dx = log|x +√(x2 – a2)| + C
- ∫ √(x2 – a2).dx =1/2.x.√(x2 – a2)-a2/2 log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2).dx = sin-1 x/a + C
- ∫1/(x2 – a2).dx = 1/2a.log|(x – a)(x + a| + C
- ∫ 1/(a2 – x2).dx =1/2a.log|(a + x)(a – x)| + C
- ∫1/√(x2 + a2 ).dx = log|x + √(x2 + a2)| + C
- ∫√(a2 – x2).dx = 1/2.x.√(a2 – x2).dx + a2/2.sin-1 x/a + C
Integration Formulas- Application
There are two sorts of integrals in general. They are integrals that are either definite or indefinite..
Definite Integration Formula
These are integrations with a pre-existing value of limits, resulting in a determined end value of integral.
= G(b) – G(a)
Indefinite Integration Formula
These are integrations that do not have a pre-existing limit value, rendering the integral’s final value indefinite. The integration constant C is used here. g'(x) = g(x) + C
We use the integration formulas discussed so far in approximating the area bounded by curves, evaluating average distance, velocity, and acceleration oriented problems, finding the average value of a function, approximating the volume and surface area of solids, estimating the arc length, and finding the kinetic energy of a moving object using improper integrals.
Integration Formulas Examples
Example 1: Find the value of ∫ (9x+ 25)/ (x+3)2 .dx
(9x+ 25)/ (x+3)2 is a rational function.
Using the partial fraction decomposition, we have (9x+ 25)/(x+3)2 = A/(x+3) + B/(x+3)2
Taking LCD, we get
(9x+ 25)/(x+3)2 = [A(x+3) +B]/(x+3)2
Equating the numerator, we get
9x+ 25 = A(x+3) +B
Solving for B when x = -3, we get B = -2
Solving for A when x = 0, we get A = 9
Thus the partial fraction is decomposed as 9/(x+3) -2/(x+3)2
As stated in the integration formulas above, find the integral of 9/(x+3) -2/(x+3)2 .
∫[9/(x+3)]dx – ∫ -2(x+3)2 .dx = 9 ln(x+3) – 2 /(x+3) +C
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Integration Formulas- QNAs
What is integration with example?
Integration is described as bringing previously isolated objects or people together.
When schools were desegregated and there were no longer separate public schools for African Americans, this was an example of integration.
What does integration mean in math?
In mathematics, integration is the process of finding a function g(x) whose derivative, Dg(x), is equal to a given function f. (x).
This is represented by the integral symbol “∫” as in f(x), which is commonly referred to as the function’s indefinite integral.
How many integration formulas are there?
There are three different types of integration methods, each with its own set of algorithms for calculating integrals. They are the outcomes that have been standardised. Integration formulas are a good way to memorise them.