1. Indefinite integrals
Definition: if f(x) be any differentiable function of such that d/dx f( x ) = f(x)Is called an anti-derivative or an indefinite
integral of f(x) we write f(x) = ∫ f (x) dx. The symbol
∫ dx is
used as a symbol of operation which means integrating
w.r.t. x.

2. 1.∫ k f(x) dx = k ∫ f(x) dx, where k is a constant
In all the above formulas, c is the constant of integration.
Some standard results on integration
1.∫ k f(x) dx = k ∫ f(x) dx, where k is a constant
2. ∫ ( f(x) ± g(x))dx = ∫ f(x)dx ± ∫ g(x)dx

3. Example:
∫4x⁷ dx
=4∫x⁷ dx
=4∫xⁿ⁺¹/n+1
=4 x⁸/ + c
= x⁸/8 + c 8

4. (1) ∫(sinx + cosx) dx
= ∫sinx dx + ∫cosx dx
= -cosx + sinx + C

5. METHODS OF INTEGRATION
It was based on inspection, i.e,. On the search
of a function F whose derivative is f which led
us to the integral of f . However, this method, which
depends on inspection , is not very suitable for
many functions.
Hence, we need to develop additional techniques
or methods for finding the integrals by reducing
Them into standard forms.

6. Following are the method
of integration:
Integration by substitution
Integration by part
Integration by partial fractions

7. Integration by substitution
This method is used when basic rules of
Integration given above, are not directly
applicable. In such cases, the integration
is transformed (by suitable substitution
of given variable by a new variable) into a
form to which basic rules of integration are
applicable

8. ∫√t dt = ∫t½ dt = 2/3t³⁄² + C Equation of integrals √15 + logx dx x
put
15 + logx = t
1/x dx = dt
Substituting in given question, we get
∫√t dt = ∫t½ dt = 2/3t³⁄² + C

9. Conti……
= 2/3 (15 + logx)³⁄² + C
Where c = Constant of integration.

10. ∫ Integration by parts ∫ ∫ ∫ This method is based on product rule
differentiation.
According to this method, the integral
of product of Two functions
f(x). g(x) is given by ∫
g(x) dx - f ’(x) ( g(x)dx) dx ∫ ∫ ∫
F(x) . g(x) dx = f(x)

11. ∫ I . II dx = I ∫IIdx - ∫(d(I)/dx) dx
Taking f(x) as the first function (I) and
g(x) as the second (II) the above equation
can be Stated as
∫ I . II dx = I ∫IIdx - ∫(d(I)/dx) dx

12. Rule for the proper chose of the first function
The first function to be selected may be the one which comes first in the order ILATE where
I means inverse trigonometric function
L means logarithmic function
A means algebraic
T means trigonometric
E means exponential.

13. ∫x² e dx x x
Take x² as the function and e as the second function and integrating by parts, we get
I = x² e - ∫2x e x X dx

14. I = x² e – 2 [xe - ∫e dx] = x² e – 2xe + 2e + C,
Again apply by parts in second term, we get x x
I = x² e – 2 [xe - ∫e dx] X X
= x² e – 2xe + 2e + C,
Where c. is the constant of integration X

15. Integration by partial fraction
If f(x) and g(x) are two polynomials,
then f(x)/g(x) is called a rational algebraic
Function of x.
Any rational f (x)/ g(x) can be expressed as
the sum of rational functions each having a
simple factor of g(x). Each such fraction is
called a partial fraction.

16. 3x
1. ∫ dx
(x-1) (x+2) A B = +
X-1
X + 2
3x = A (X + 2) + B (X + 1)

17. Put,,
X = we get
3 = 3A
A = 1
Put,
x = -2 we get
- 6 = - 3B
B = 2

18. ∫ ∫ ∫ = (x+2) (x-1) 2 + 1 X - 1 3x X-1 X + 2 3x dx + 2 X + 2
Now integrating, we get 3x 2 1
X - 1 ∫ ∫
dx = ∫
dx + 2
( x - 1 ) ( x + 2)
X + 2

19. = log ( x – 1 ) + 2 log ( x + 2 ) + c

20. Thank you
Presented by,
NUZHAT SHAHEEN