There are four integrals:
1) definite integral x/(1+x^4)dx b/w 0_infinity
2) definite integral (x^2)/(1+x^4)dx b/w 0_infinity
3) definite integral (x^3)/(1+x^4)dx b/w 0_infinity
4) definite integral (x^4)/(1+x^4)dx b/w 0_infinity
Which of these integrals converge. First of all, what does it mean "converge"? How do you compare to "pure" powers of x?
How would you compute the exact value of at least one of the convergent integrals?
thanks
To understand what it means for an integral to converge, let's start with the concept of convergence. In mathematics, convergence refers to the behavior of a sequence or a series as its terms approach a certain value or limit. In the context of integrals, convergence means that the integral is finite and does not approach infinity as the bounds of integration become larger.
To compare these integrals to "pure" powers of x, we can consider their behavior as x approaches infinity. If the integrals behave similarly to a "pure" power of x, they will have the same convergence properties as that power. By comparing the exponents of x and the integrals, we can determine their convergence.
Now, let's analyze each integral separately:
1) Integral of x/(1+x^4)dx from 0 to infinity:
To determine convergence, we consider the exponent of x in the numerator and denominator. Here, the highest power of x is 1 in both the numerator and denominator. Since the exponents are the same, we need to evaluate the limit as x approaches infinity. In this case, the integral converges because the highest power of x appears in the denominator, resulting in the integral approaching 0 as x tends to infinity.
2) Integral of (x^2)/(1+x^4)dx from 0 to infinity:
The numerator here has a higher power of x (2) compared to the denominator (4). This suggests that the integral might converge since the numerator grows at a slower rate than the denominator. To confirm this, we evaluate the limit as x approaches infinity. By evaluating this limit, we find that the integral converges.
3) Integral of (x^3)/(1+x^4)dx from 0 to infinity:
Similarly, here the numerator has a higher power of x (3) compared to the denominator (4). This suggests possible convergence. Evaluating the limit as x approaches infinity, we find that the integral also converges.
4) Integral of (x^4)/(1+x^4)dx from 0 to infinity:
In this case, the powers of x in the numerator and denominator are the same (4). This suggests that the integral may not converge as x approaches infinity. To confirm this, we evaluate the limit as x approaches infinity. By evaluating this limit, we find that the integral diverges.
Therefore, the first three integrals (1, 2, and 3) converge, while the fourth integral (4) diverges.
To compute the exact value of one of the convergent integrals, let's consider the first integral: Integral of x/(1+x^4)dx from 0 to infinity.
To solve this integral, we can use the method of residues from complex analysis. First, we recognize that the integrand is an odd function, so we can split the integral into two parts: the integral from 0 to infinity and the integral from negative infinity to 0. Then, we can convert the integral from negative infinity to 0 into a contour integral in the complex plane.
By using the techniques of complex analysis and integrating over a semicircular contour in the upper half-plane, we can find the residues and evaluate the integral. The final result is π/2.
Note that the process of finding the exact value of an integral can vary depending on the specific integral and its properties. In this case, the method of residues is applicable, but other integrals may require different techniques like integration by parts or trigonometric substitutions.
To determine whether an integral converges, we need to check if the integral value exists and is finite. In other words, if the integral has a well-defined value.
In the context of comparing to "pure" powers of x, we can consider the behavior of the integrand as x approaches infinity. If the function in the integrand approaches zero as x approaches infinity faster than the corresponding power of x, then the integral converges.
Now, let's analyze each integral and determine which ones converge:
1) Definite integral of x/(1+x^4)dx from 0 to infinity:
To determine convergence, let's consider the limit as x approaches infinity of the integrand:
lim(x->∞) [x/(1+x^4)]
As x approaches infinity, the denominator, 1+x^4, becomes negligible compared to x. Therefore, the integrand approaches 1/x, which is a pure power of x.
Since the integrand resembles a pure power of x, this integral does not converge.
2) Definite integral of (x^2)/(1+x^4)dx from 0 to infinity:
Similar to the first example, let's examine the limit as x approaches infinity:
lim(x->∞) [(x^2)/(1+x^4)]
As x goes to infinity, both the numerator and denominator grow without bound. However, the denominator grows faster since it involves a fourth power of x. Therefore, the integrand approaches zero, indicating convergence.
3) Definite integral of (x^3)/(1+x^4)dx from 0 to infinity:
Again, considering the limit as x approaches infinity:
lim(x->∞) [(x^3)/(1+x^4)]
Similar to the previous example, the numerator grows faster than the denominator, resulting in the integrand approaching infinity as x goes to infinity. Hence, this integral does not converge.
4) Definite integral of (x^4)/(1+x^4)dx from 0 to infinity:
One more time, let's evaluate the limit as x approaches infinity:
lim(x->∞) [(x^4)/(1+x^4)]
In this case, both the numerator and denominator grow at the same rate. As a result, the integrand approaches a finite value, indicating convergence.
To compute the exact value of one of the convergent integrals, let's focus on integral number 2:
Definite integral of (x^2)/(1+x^4)dx from 0 to infinity:
Let's make the substitution u = x^2, which transforms the integral into a more manageable form:
∫(0 to infinity) [2u/(1+u^2)] du
To evaluate this integral, we can use partial fractions or trigonometric substitutions, depending on the approach you choose. Would you like to proceed with either of these methods?