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
examine the powers.
4) x^4 dx is an x^5 in the numerator, an x^4 in the denominator. It cant converge to a finite value at inf.
1,2) definitely do, they converge to zero at inf
3) Think on that.
do you do u substitution for each or automatically trig substitution. Because when you do trig sub you get 0 but when you do u sub then you get pi/4
In the context of integrals, "convergence" refers to whether the value of the integral exists or is finite. If an integral converges, it means that the area under the curve is finite, and the integral can be evaluated to a specific value. If an integral diverges, it means that the area under the curve is infinite, and the integral does not have a well-defined value.
To determine if these integrals converge or diverge, we can compare them to the "pure" powers of x. A "pure" power of x refers to integrals of the form ∫(x^n)dx, where n is a constant exponent. The convergence of pure power integrals depends on the value of n.
Let's compare each of the given integrals with the pure powers of x:
1) The integral ∫(x/(1+x^4))dx can be compared with the integral of x^1, as n = 1. We can check if the power x^1 dominates over the function (1+x^4) or vice versa as x approaches infinity. If x dominates, the integral will converge, and if (1+x^4) dominates, the integral will diverge.
2) The integral ∫(x^2/(1+x^4))dx can be compared with the integral of x^2, as n = 2. Similar to the previous case, we need to compare if x^2 dominates over (1+x^4) or vice versa as x approaches infinity.
3) The integral ∫(x^3/(1+x^4))dx can be compared with the integral of x^3, as n = 3. We compare the dominance of x^3 and (1+x^4) as x approaches infinity.
4) The integral ∫(x^4/(1+x^4))dx can be compared with the integral of x^4, as n = 4.
To compute the exact value of at least one of the convergent integrals, let's consider the first integral ∫(x/(1+x^4))dx:
One way to compute this integral is to use a technique called partial fractions. We can express the integrand as a sum of simpler fractions:
x/(1+x^4) = (1/4) * [(x / (x^2 + √2x + 1)) + (x / (x^2 - √2x + 1))]
Using partial fractions, we can solve for the coefficients and rewrite the integral in terms of simpler fractions. Then we can evaluate each integral and obtain an exact value, given the limits of integration.
Please note that the computations for the other integrals may involve different techniques or methods, depending on the respective convergence or divergence of each integral.