integral from 1 to infinity 4/(x^2)(x^2+1)dx
To evaluate the integral on this question, we will use a method known as partial fraction decomposition. Here are the step-by-step instructions to solve it:
Step 1: Write the integrand as a sum of partial fractions.
Let's start by factoring the denominator:
x^2(x^2 + 1) = x^2(x + i)(x - i)
Now, our partial fraction decomposition will have the following form:
4/(x^2)(x^2 + 1) = A/x + B/x^2 + (Cx + D)/(x^2 + 1)
Step 2: Find the values of A, B, C, and D.
To determine the values of the constants A, B, C, and D, we need to find a common denominator and equate the numerators:
Multiply both sides of the equation by x^2(x^2 + 1):
4 = A(x^2 + 1) + B(x)(x + i) + (Cx + D)(x^2)
Expanding and grouping like terms:
4 = A(x^2 + 1) + B(x^2 + ix) + Cx^3 + Dx^2
Now, compare the coefficients of the powers of x on both sides of the equation:
For the constant term: 0 = A + B
For the term with x: 0 = B + C
For the term with x^2: 0 = A + D
For the term with x^3: 0 = C
From these equations, we can determine the values of A, B, C, and D:
A = -D
B = -C
0 = C (which means C = 0)
0 = D (which means D = 0)
A = 0
B = 0
Therefore, the partial fraction decomposition of 4/(x^2)(x^2 + 1) is:
4/(x^2)(x^2 + 1) = 0/x + 0/x^2 + 0/(x^2 + 1)
Step 3: Evaluate the integral using the partial fractions.
Now that we have the partial fractions, we can integrate each term separately:
∫[4/(x^2)(x^2 + 1)]dx = ∫[0/x]dx + ∫[0/x^2]dx + ∫[0/(x^2 + 1)]dx
The integral of 0/x and 0/x^2 is straightforward and equal to 0. Thus, we're left with:
∫[0/(x^2 + 1)]dx
To evaluate this integral, we can use a basic trigonometric substitution:
Let x = tan(theta), which implies dx = sec^2(theta) d(theta)
Substituting these values back into the integral:
∫[0/(x^2 + 1)]dx = ∫[0/(tan^2(theta) + 1)]sec^2(theta) d(theta)
Simplifying the integrand:
∫[0/(tan^2(theta) + 1)]sec^2(theta) d(theta) = ∫[0/((sin^2(theta)/cos^2(theta)) + 1)]sec^2(theta) d(theta)
Combining the terms and simplifying:
∫[0/((sin^2(theta) + cos^2(theta))/cos^2(theta))]sec^2(theta) d(theta) = ∫[0/(1/cos^2(theta))]sec^2(theta) d(theta)
Further simplifying:
∫[0/(1)]sec^2(theta) d(theta)
The integral on the right-hand side is now a basic integral that we can evaluate easily:
∫[0/(1)]sec^2(theta) d(theta) = tan(theta) + C
Step 4: Convert back to the original variable.
Finally, we need to convert back to the original variable, x. Recall that we set x = tan(theta):
Therefore, our final answer is:
∫[4/(x^2)(x^2 + 1)]dx = tan(theta) + C
Remember, tan(theta) = x, so the final answer is:
∫[4/(x^2)(x^2 + 1)]dx = x + C
In conclusion, the integral from 1 to infinity of 4/(x^2)(x^2 + 1)dx is x evaluated from 1 to infinity, which means the integral diverges.