Q.Find: $\displaystyle\int \dfrac{x^2}{(x^2 + 4)(x^2 + 9)}\, dx$

The key idea is Partial Fraction Decomposition for rational functions, simplified by a temporary substitution.

First, we observe the integrand has $x^2$ terms in the denominator. Let $y = x^2$ to simplify the partial fraction decomposition:

Multiplying by $(y+4)(y+9)$ gives $y = A(y+9) + B(y+4)$.

Setting $y = -4$, we get $-4 = A(-4+9) \implies -4 = 5A \implies A = -\frac{4}{5}$.

Setting $y = -9$, we get $-9 = B(-9+4) \implies -9 = -5B \implies B = \frac{9}{5}$.

So, the expression becomes $\frac{9}{5(y+9)} - \frac{4}{5(y+4)}$.

Substituting back $y=x^2$, the integral is:

We use the standard integral $\int \frac{1}{x^2+a^2}\, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$: …

The integrand is a rational function, $\frac{x^2}{(x^2+4)(x^2+9)}$. Notice a key structural feature: the variable $x$ only appears as $x^2$. This suggests a simplification strategy for the partial fraction decomposition step.

Concept and Intuition:

When dealing with rational functions where the variable appears consistently as $x^2$ (e.g., $\frac{P(x^2)}{Q(x^2)}$), we can simplify the algebraic process of partial fraction decomposition by temporarily treating $x^2$ as a single variable. This is not a formal substitution for integration (like a $u$-substitution), but rather an algebraic trick to make the decomposition easier.

Let $y = x^2$. The expression then becomes $\frac{y}{(y+4)(y+9)}$. This is a standard rational function in $y$ that can be decomposed into partial fractions of the form $\frac{A}{y+4} + \frac{B}{y+9}$. Once we find the constants $A$ and $B$, we substitute $x^2$ back for $y$. This transforms the original integrand into a sum of terms like $\frac{A}{x^2+4}$ and $\frac{B}{x^2+9}$. These are standard integrals that evaluate to inverse tangent functions.

Step-by-step solution:

1. Algebraic Substitution for Partial Fraction Decomposition: Let $y = x^2$. The integrand can be rewritten as:

We will decompose this expression into partial fractions. Since the denominator has two distinct linear factors in $y$, the decomposition will be of the form:

To find the constants $A$ and $B$, we combine the terms on the right side:

Equating the numerators, we get the identity:

2. Determine the Constants $A$ and $B$:

   We can find $A$ and $B$ by substituting convenient values for $y$ into the identity $y = A(y+9) + B(y+4)$.

   • To find $A$, set $y = -4$:

*   To find $B$, set $y = -9$:

Thus, the partial fraction decomposition for the algebraic expression in $y$ is:

3. Substitute back $x^2$ for $y$: Now, replace $y$ with $x^2$ in the decomposed expression to get the form suitable for integration:

It's often clearer to write the positive term first:

4. Integrate the Decomposed Terms: We now need to evaluate the integral of this sum: $\int \frac{x^2}{(x^2+4)(x^2+9)}\, dx = \int \left( \frac{9}{5(x^2+9)} - \frac{4}{5(x^2+4)} \right)\, dx$ …