Integral Numbers

Integration is an important concept in mathematics and, together with differentiation, is one of the two main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral

a

∫    f(x) dx

b

is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.

-Source Wikipedia

Types of Integral numbers Methods:

The integral numbers different types are given as follow:

Substitution
Partial Fractions
Parts

1) Integration by substitution for integral numbers methods:

The particular integral ∫ f(z) dz can be converted into a dissimilar form by substituting the independent variable y to t,

Let assume J = ∫ f(z) dz

Put y = h(t) , so dz/dt = h’(t) we write dz = h’(t) dt .

Thus J = ∫ f(z) dz = ∫ f(h(t)) h’(t) dt.

Example: Integrate cos nz.

Solution:

nz = t

n dz = dt (or) dz = dt / n.

Therefore, ∫ cos nz dz = ∫ cos t dt/n

= 1/n ∫ cos t dt

= 1/n sin t + c

Here: ∫ cos z dz = sinz + D

= 1/n sin nz + D to change t as nz.

The above 2  methods are used to decrease the standard form of the particular equations.

Integration using Partial fractions for integral numbers method:

A rational function is explained as the ratio of 2  polynomials in the form. Q(z) / R(z) , here Q(z) and R(z) are polynomials in Z. Here R(z) ≠ 0.

Example:

Integrate ∫ dz / (z+3) (z+4)

Solution:

Here, the integrand is a proper rational function.

We write, 1 / (z+3) (z+4) = C/z+3 + D/z+4

where, real numbers C and D are to be calculated by cross multiplication, it gives,

1 = C(z+4) + d(z+3).

Applying substitute z as -3, we get

C = 1

Applying substitute z as -4, we get

B = -1

Thus, 1/ (y+3)(y+4) = 1/(y+3) + -1/(y+4)

Therefore, take integral

∫ dz / (z+3)(z+4) = ∫ dz / z+3 - ∫ dz / z+4

= log │z+3│- log │z+4│+ D

Here: ∫ dz / z+c = log │z+c│

= log │z+3/z+4│+ d