IMPROPER INTEGRALS OF FIRST AND SECOND KIND

Integrals with infinite limits of integration are called improper integrals of integration of type I.

where c is a real number

Integrals of functions that become infinite at a point within the interval of integration are called improper integrals of Type II

In each case, if the limit is finite set we sat that the improper integral converges and that the limit is the value of the improper integral.

If the limit fails to exists, the improper integral diverges.

Problem 1 :

Solution :

Let us consider the given function as f(x),

Here f(x) is continuous on [a, ∞), then

So, the integral converges and equal to 1.

Problem 2 :

Solution :

In the function f(x), the denominator x² - 4 will become 0 when x = 2. So, the function is not defined at x = 2.

f(x) is continuous on [a, b) and discontinuous at x = b

So, the integral diverges.

Problem 3 :

Solution :

1∈(0, 3), at x = 1 the function f(x) is not defined. So, we need split it into two integrals.

Problem 4 :

Solution :

f(x) is continuous on [a, ∞), then

Problem 5 :

Solution :

Let u = ln x, du = (1/x) dx

v = 1/x², dv = -1/x

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