Determine whether the integral is convergent or divergent.If it is convergent, evaluate it.
form -infinity to 0
x/(x^4+25)dx
To evaluate
∫x/(x^4+25) dx
let
z^2 = x
2z dz = dx
now you have
1/2 ∫1/(z^2+25) dz
That is just a standard integral, giving
(1/2)(1/5)arctan(z/5)
evaluate at the limits and you get -π/20
Well, let me take a wild guess and assume that the integral is convergent. After all, life is full of surprises and unexpected convergences!
To evaluate it, we can use a clever trick. Let's factor out an "x" from the numerator and denominator:
∫(x/(x^4+25)) dx = ∫(x/(x^4(1 + (5/x^4)))) dx
Now, if we let u = x^4, then du = 4x^3 dx. Rearranging, we get dx = du/(4x^3).
Substituting these into the integral, we have:
∫(x/(x^4+25)) dx = ∫(1/(4u(1 + 5/u))) du
= ∫(1/(4u + 5)) du
To integrate this, we can use a simple substitution. Let v = 4u + 5. Then dv = 4du, so du = dv/4.
Substituting back, we have:
∫(1/(4u + 5)) du = ∫(1/v) (dv/4)
= (1/4) ln|v| + C
Remembering that v = 4u + 5, we can substitute back again:
(1/4) ln|v| + C = (1/4) ln|4u + 5| + C
Finally, substituting back for u = x^4, we get:
(1/4) ln|4x^4 + 5| + C
So, the integral is convergent and its value is (1/4) ln|4x^4 + 5| + C.
Now, I've done my part. It's up to you to plug in the relevant values for evaluation. But hey, remember to take the absolute value of the expression inside the natural logarithm, or you might end up in the land of complex numbers!
To determine whether the integral ∫(from -∞ to 0) x/(x^4+25) dx is convergent or divergent, we can use the limit comparison test.
Let's find a function g(x) whose integral is known and easier to evaluate. We can choose g(x) = 1/x^2 as our test function.
Now, let's consider the function f(x) = x/(x^4+25). To use the limit comparison test, we need to compare the behavior of f(x) and g(x) as x approaches -∞.
Taking the limit as x approaches -∞:
lim(x->-∞) (f(x) / g(x))
= lim(x->-∞) (x/(x^4+25) / (1/x^2))
= lim(x->-∞) (x^3 / (x^4+25))
To simplify the expression, we can divide both the numerator and denominator by x^4:
= lim(x->-∞) (1 / (1/x^4 + 25/x^4))
= lim(x->-∞) (1 / (1 + 25/x^4))
= 1
Since the limit is a finite non-zero value, the integral is convergent.
Now, let's evaluate the integral:
∫(from -∞ to 0) x/(x^4+25) dx
To do this, we can make a u-substitution. Let u = x^2, so du = 2x dx.
When x = 0, u = 0. When x = -∞, u = ∞.
Now, let's rewrite the integral using u:
∫(from u=0 to ∞) (1/2) * (1/(u^2+25)) du
This integral can be evaluated as:
(1/2) * arctan(u/5) (evaluated from u=0 to ∞)
Plugging in the limits:
= (1/2) * (arctan(∞/5) - arctan(0/5))
= (1/2) * (π/2 - 0)
= π/4
Therefore, the integral ∫(from -∞ to 0) x/(x^4+25) dx is convergent and its value is π/4.
To determine whether the given integral is convergent or divergent, you need to analyze the behavior of the integrand as the variable approaches the limits of integration.
Step 1: Analyze the integrand at the limits
In this case, the limits of integration are from negative infinity to 0. Let's substitute the limits into the integrand:
- At x = -∞ (negative infinity), the integrand becomes:
lim x→-∞ x/(x^4 + 25) = -∞/∞
This limit is indeterminate and inconclusive.
- At x = 0, the integrand becomes:
x/(x^4 + 25) = 0/(0^4 + 25) = 0/25 = 0
Step 2: Simplify the integrand
To analyze the integrand further, you can simplify it by dividing both the numerator and the denominator by the highest power of x in the denominator, which is x^4.
Dividing the numerator and the denominator by x^4, we get:
x/(x^4 + 25) = x/(x^4(1 + 25/x^4)) = 1/(x^3 + 25/x)
Step 3: Analyze the simplified integrand
Now, let's analyze the behavior of the simplified integrand as x approaches -∞ and 0.
- As x approaches -∞, the denominator x^3 approaches -∞, which means the value of the simplified integrand approaches 0. Hence, the integrand is finite and convergent at -∞.
- As x approaches 0, the denominator 25/x approaches ∞, which means the value of the simplified integrand approaches 0. Hence, the integrand is finite and convergent at x = 0.
Step 4: Evaluate the integral (if convergent)
Since the integrand is convergent at both limits, we can evaluate the integral.
To evaluate the integral ∫(x/(x^4 + 25))dx from -∞ to 0, we can use the definite integral:
∫[a to b] f(x)dx = F(b) - F(a)
where F(x) is the antiderivative of f(x).
By finding the antiderivative F(x), we have:
F(x) = ∫(1/(x^3 + 25/x))dx
Now, you can find the antiderivative using various techniques such as u-substitution, trigonometric substitution, or partial fractions. Once the antiderivative is determined, substitute the limits of integration to evaluate the integral.
Unfortunately, due to the complexity of this integral, it is not possible to provide the exact evaluation without further calculations.