1. Integral of a function, integrand
If the derivative of a function f(x) is F(x) i.e. if
we say that f(x) is an integral or primitive of F(x) and in symbols, we write
The letter x in dx denotes that the integration is to be performed with respect to the variable x.
2. Integral as an anti-derivative
The process of determining an integral of a function is called integration and the function to be integrated is called the integrand.
Since integration and differentiation are inverse processes we have
3. Indefinite Integration, General Integral, Arbitrary Constant
If f(x) is an integral of F(x) then f(x)+c is also an integral of F(x), c being a constant what so ever for
where c is an arbitrary constant. If f(x) be an integral of F(x), then f(x)+c is its general integral.
As c is indefinite in nature, that’s why such integration is known as indefinite integration.
4. Theorems on Indefinite Integration
∫af(x)dx=a∫f(x)dx where a is a constant
∫(f(x)±g(x))dx=∫f(x)dx ± ∫g(x)dx
5. Fundamental Integrals
6. Integration by substitution
In this method the integral ∫f(x)dx is expressed in terms of another integral where some other variables say t is the independent variable; x and t being connected by some suitable relation x=g(t) . It leads to the result ∫f(x)dx =∫f(g(t)). g' (t) dt
7. Standard Substitutions
(a). For terms of the form x2 +a2 or (x2 +a2 )1/2 put x=a tan t
(b). For terms of the form x2 - a2 or (x2 - a2 )1/2 put x=a sec t
(c). For terms of the form a2 - x2 or (a2 - x2 )1/2 put x=a sin t
(d). When both (a + x )1/2 (a - x )1/2 are present put x=a cos 2t
(e). For the terms of the form ((x-a)(b-x))1/2 put x=a cos2 t + b sin2 t
(f). For terms of the form (x±(x2 - a2 )1/2 )n put t=x±(x2 - a2 )1/2
8. Integration by parts
If u,v be two differential functions of x. We have
Integrating both sides we get,
Let u=f(x) and
So, ∫f(x) g(x) dx = f(x) ∫g(x)dx - ∫ [(∫g(x)dx ). f' (x)] dx
In words, this states that,
The integral of (product of two functions) =first function*integral of the second-integral of (derivative of first function*integral of second function)
The first and second function is decided by the ILATE rule where
9. Integration by Partial Fractions
If the integral is in the form of an algebraic fraction which cannot be integrated then the fraction needs to be decomposed into partial fractions.
Rules for expressing in partial fraction:
1. The numerator must be at least one degree less than the denominator.
2. For every factor (ax+b) in the denominator, there is a partial fraction
3. If a factor is repeated in the denominator n times then that partial fraction should be written n times with degree 1 through n
4. For a factor of the form ( ax2 + bx +c )in the denominator, there will be a partial fraction of the form
10. Integration using trigonometric functions
Integrations of the form ∫sinmx cosnx can be solved for the following cases
can be solved for the following cases
Case 1: Either m or n is odd
Write all even powers of sin x in terms of cos x
For the remaining sin x write sinx dx as - d(cos x)
Case 2: When both m and n are even
Use double angle formula
Case 3: When m and n are not all positive integers
Convert sin x and cos x in terms of sec x and tan x
(i). If the power of sec x is even, use substitution u=tan x ⇒ du=sec2x dx
Use 1+tan2x = sec2x
(ii). If the power of tan x is negative odd, use substitution with u = sec x, du = sec x tan x dx, and convert remaining powers of tan to a function of u using tan2x = sec2x - 1 . This works if m ≥ 1.
(iii). If the power of sec x is odd and tan x is even, use 1+tan2x = sec2x and convert all sec x into tan x
11. Special Integrals
Indefinite Integration for JEE Main, Download PDF!!!
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