Integration of x^2/(x^4 * x^2 + 1)dx

To integrate the given expression, we can use the method of substitution. Let's consider the substitution u = x^2. This implies du/dx = 2x, and solving for dx gives dx = du/(2x).

Now, let's rewrite the integral using this substitution:

∫(x^2/(x^4 * x^2 + 1))dx = ∫(u/(u^2 * x^2 + 1))(du/(2x))

Simplifying the expression, we have:

∫(u/(u^2 * x^2 + 1))(du/(2x)) = (1/2)∫(u/(u^2 * x^2 + 1))(du/x)

Now, let's focus on the term (u^2 * x^2 + 1). We can rewrite it as follows:

u^2 * x^2 + 1 = u^2 * x^2 + u^0
= u^2 * x^2 + u^2 * 1^2
= u^2(x^2 + 1)

Substituting this back into the integral, we have:

(1/2)∫(u/(u^2 * x^2 + 1))(du/x) = (1/2)∫(u/(u^2(x^2 + 1)))(du/x)

Now, we can cancel out the common factor of x in the numerator and denominator:

(1/2)∫(u/(u^2(x^2 + 1)))(du/x) = (1/2)∫(u/(u^2(x^2 + 1)))(du)

Next, we can split the fraction into two separate fractions:

(1/2)∫(u/(u^2(x^2 + 1)))(du) = (1/2)∫(1/(x^2 + 1))(du/u) + (1/2)∫(1/(u(x^2 + 1)))(du)

The first integral, (1/2)∫(1/(x^2 + 1))(du/u), can be evaluated as the natural logarithm of the absolute value of u:

(1/2)∫(1/(x^2 + 1))(du/u) = (1/2)ln|u| + C1

Where C1 is the constant of integration.

Similarly, the second integral, (1/2)∫(1/(u(x^2 + 1)))(du), can be evaluated as the natural logarithm of the absolute value of u^2(x^2 + 1):

(1/2)∫(1/(u(x^2 + 1)))(du) = (1/2)ln|u^2(x^2 + 1)| + C2

Where C2 is the constant of integration.

Now, substituting back u = x^2, we have:

(1/2)ln|u| + C1 = (1/2)ln|x^2| + C1 = (1/2)ln(x^2) + C1 = ln|x|