Evaluating the Integral of x^2 / ((x^4+x^2+1)(ax^2+1))

Published on August 11, 2026

Calculus

Learn how to evaluate the complex rational integral using partial fraction decomposition and algebraic substitution techniques.

Question

integral ((x^2/(x^4+x^2+1)(ax^2+1))

Answer

To evaluate the integral

we can use a substitution method by manipulating the integrand to create a form suitable for integration.

Step 1: Algebraic Manipulation

First, let’s divide the numerator and denominator by

Notice that

. Let

. Then

. However, the term

makes this direct substitution difficult.

Instead, let’s use partial fraction decomposition on the term

.
Let

. We are looking for a decomposition of the form:

Multiplying through by the denominator:

Equating coefficients of

Substitute

into the second equation:

Then:

Step 2: Rewriting the Integral

The integral becomes:

Let

Part A:

Part B:
Perform polynomial division:

(Wait, let’s re-divide):

So,

Step 3: Combining and Simplifying

Notice that the

from Part B cancels the

from Part A.

To solve

, divide by

This is generally solved by splitting the numerator into

and

.
Let

The remaining part

is solved by

Final Answer Form:

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