Find out what kind of improper integral is given below
§dx/{x^4+4}....?,hence evaluate the integral with
upper boundary=infinity
lower boundary =0
i need help plz show me full work
let x = 2tanθ
x^2+4 = 4tan^2θ+4 = 4sec^2θ
dx = 2sec^2θ dθ
now the integrand becomes
2sec^2θ/4sec^2θ dθ = 1/2 dθ
Now recall that tan(π/2) = ∞
This is just a reminder of one of the standard forms:
∫ dx/(x^2+a^2) = 1/a arctan(x/a)
so what do i do from here now sir
Oh. I'm sorry. I misread the integral. x^4+4 is a bit trickier.
But, as usual, google turned up several discussions. You might care to start here:
http://www.freemathhelp.com/forum/threads/55678-difficult-integration-int-1-(1-x-4)-dx
If you follow the threads of that web site, you will probably see how wolframalpha came up with its soution:
http://www.wolframalpha.com/input/?i=%E2%88%AB%5B0,infinity%5D+dx%2F(x%5E4%2B4)
To determine the kind of improper integral, we need to investigate the behavior of the integrand as the upper boundary approaches infinity.
The given integral is:
∫(0 to ∞) dx / (x^4 + 4)
To evaluate this integral, we can start by identifying the kind of improper integral using the following steps:
Step 1: Check for existence issues
Let's examine if the denominator satisfies any of the following conditions:
1. The denominator becomes zero at a point within the interval of integration.
2. The denominator becomes infinite within the interval of integration.
In this case, for the denominator x^4 + 4, there are no real values of x that make the denominator zero, and it does not become infinite within the interval of integration.
Step 2: Determine the endpoint limits
Evaluate the integral at each individual endpoint (0 and ∞). We will consider each endpoint separately.
a) Endpoint: x = 0
Substituting 0 into the integral, we have:
∫(0 to ∞) dx / (x^4 + 4) = ∫(0 to ∞) dx / (0^4 + 4) = ∫(0 to ∞) dx / 4 = `[dx]_0^∞ / 4`
b) Endpoint: x = ∞
To determine the behavior of the integrand as x approaches infinity, we can use limits. Let's calculate the limit as x approaches infinity:
lim(x→∞) dx / (x^4 + 4)
Dividing both the numerator and denominator by x^4, we get:
lim(x→∞) (1/x^4) dx / (1 + 4/x^4)
As x approaches infinity, the second term in the denominator, 4/x^4, approaches 0. Therefore, we can ignore it in the limit:
lim(x→∞) (1/x^4) dx / (1 + 0)
This simplifies to:
lim(x→∞) (1/x^4) dx / 1 = lim(x→∞) 1/x^4 dx
Integrating this limit, we have:
∫(∞ to ∞) 1/x^4 dx = [(-1/3x^3)]_∞^∞ = 0
Step 3: Determine the kind of improper integral
Now, combining the results from Step 2:
The integral ∫(0 to ∞) dx / (x^4 + 4) diverges, as both endpoint limits do not exist or do not converge to a finite value. Therefore, it is an improper integral of the first kind.
In summary, the given integral ∫(0 to ∞) dx / (x^4 + 4) is an improper integral of the first kind and it diverges.
Please note that evaluating the value of this improper integral is not possible as it diverges.