Let F(x) be a differentiable function of x such that d/dx [F(x)] = f(x). Then F(x) is called the integral of f(x). Symbolically, it is written as ∫ f(x) dx = F(x).
f(x), the function to be integrated, is called the integrand.
F(x) is also called the anti-derivative (or primitive function) of f(x).
As the differential coefficient of a constant is zero, we have
d/dx (F(x)) = f(x) ⇒ d/dx [F(x) + c] = f(x).
Therefore, ∫ f(x) dx = F(x) + c. This constant c is called the constant of integration and can take any real value.
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Some of the important integrals of some frequently used functions are listed below:
Some Standard Formulae |
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Example 1:
Evaluate ∫ 1/(1 + sin x) dx.
Solution:
In order to compute this integral, we multiply and divide it by (1-sin x) and so we have,
∫ 1/(1+ sin x) . (1 – sin x)/(1 - sin x) dx
= ∫ (1 – sin x)/(1 - sin2x) dx
= ∫ [(1 – sin x)/cos2x] dx
Now splitting it into two terms,
= ∫(1/cos2x dx – ∫ (sin x /cos2x) dx
= ∫ sec2x dx – ∫ tan x sec x dx
= tan x – sec x + c.
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Example 2:
Evaluate ∫ sin x /(1 + sin x) dx.
Solution:
∫ sin x /(1 + sin x) dx
= ∫ (sin x + 1 – 1)/(1 + sin x) dx
= ∫1 – ∫1/(1 + sin x) dx
= ∫1 – ∫(1 – sin x)/(1–sin2 x) dx
= ∫1 dx – ∫(1–sin x)/cos2 x dx
= ∫(1 – sec2 x + sec x tan x) dx
= x - tan x - sec x + C.
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Example 3:
Evaluate ∫(sin3x + cos3x)/(sin2x cos2x) dx.
Solution:
= ∫ tan x sec x dx + ∫ cot x cosec x dx
= sec x – cosec x + c.
The following chart illustrates the various methods of integration:
These methods have been explained in detail in the coming sections. Students are advised to refer the coming sections to understand the concepts. Integration by substitution can further be divided into three parts:
It is better to memorize some of the standard substitutions as they often prove helpful in solving tricky questions.
Q1. Integration is the inverse process of
(a) the coefficient of x2.
(b) anti differentiation
(c) differentiation
(d) anti derivative
Q2. The answer of an indefinite integral
(a) necessarily has a constant
(b) may or may not have a constant
(c) cannot have a constant
(d) none of these
Q3. The differential coefficient of a constant is
(a) number itself
(b) zero
(c) constant
(d) none of these
Q4. If ∫ f(x) dx = F(x) + c, then
(a) F(x) is called the primitive function of f(x).
(b) f(x) is called the primitive function of F(x).
(c) F(x) is called the derivative function of f(x).
(d) f(x) is called the root of F(x).
Q5. While solving integrals of the form x2 + a2 or √(x2 + a2), the substitution used is
(a) x = a sec θ
(b) x = a tan θ
(c) x = a cosec θ
(d) x = a sin θ
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Q5. |
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(a) |
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You may wish to refer Integration by Substitution.
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