Integrals

       
           

The Integral of any function f (z) is equal to the sum of the individual areas bounded by the function, the x-axis and the lines x = p and x = q. mathematically, ∫ f (z) dz.

The symbol ∫ f (z) dz is read as the indefinite integral of f (z) with respect to ‘z’.

The process of computing an integral is called integration. Integration is the important operation in the integral calculus. In another words integration is defined as the inverse process of differentiation and hence the evaluation of an integral is called as anti derivative. It is used in the calculation of areas and volumes of irregular shapes and solids.

We know that the number which is written in the form of 1 / a2 + b2 is known as rational integral.

Now we see some formulas of rational integrals.

 

1. ∫ a dz = az + c, where ‘c’ is arbitrary constant of integration.

2. ∫ (1/z) dz = ln |z|+ c.

3.∫ za dz = za+1/a + 1 + c (a ≠ -1).

4. ∫ [1 / (az + b)] dz = 1/a ln |az + b|.

5. ∫ 1 da / (a + x)2 = - 1 /(a + x).

6. ∫ (az + b)n dz = (az + b)n /a(n+1) + c, where the value of n ≠ -1.

 

Exponential Function:

7. ∫ ez dz = ez + c.

8. ∫ az dz = az  / ln a + c,

 

Trigonometric Functions

9. ∫ sin z dz = - cos z + c
10. ∫ cos z dz = sin z + c
11. ∫ tan z dz = - ln |cos z| + c

12. ∫ cot z dz = ln |sin z| + c

13. ∫ sec z dz = ln |sec z + tan z| + c

14. ∫ 1 / z2 + a2 dz = 1/a tan-1(z/a) + c

15. ∫ [tan(z)*sec (z)] dz = sec (z) + C 

16. ∫ [cot(z)*csc (z)]dz = -csc (z) + C 
17. ∫ [sec2 (z)] dz = tan (z) + C 
18. ∫ [csc2 (z)] dz = -cot (z) + C
 

Integrals of the form will be,

∫ [f’(z)/f(z)] dz = ln f(z) + c.

 
Integrals of the form will be,

∫ [f(z)]n f’(z) dz = [f(z)n+1/ n+1]+ c.

 

There are two types of integrals:

1.Definite integrals with upper and lower limits, and

 

pq f (z) dz

pq f (z) dz = F(q) – F(p)

 
2. Indefinite integrals,without limits

 

∫ f (z) dz

∫ f (z) dz = F(z) + c

Math Topics