Evaluate the integral: I = integral x/(x^4+3x²+2) dx
Evaluate the integral: $$ I = \int \frac{x}{x^4+3x^2+2}\,dx. $$
Explanation of Method
Factoring the quartic denominator as a product of two quadratics in $x^2$ allows partial fraction decomposition. Since only odd powers of $x$ appear in the numerator, each partial fraction term integrates directly via a $u=x^2$-style substitution to a logarithm.
- Factor the denominator.
- Decompose into partial fractions.
- Solve for the coefficients.
- Integrate each term and combine.
Step 1 · Factor the denominator
Treating $x^4+3x^2+2$ as a quadratic in $x^2$:
$$ x^4+3x^2+2 = (x^2+1)(x^2+2). $$
Thus, the integral becomes:
$$ I = \int \frac{x}{(x^2+1)(x^2+2)}\,dx. $$
Step 2 · Partial fraction decomposition
Assume:
$$ \frac{x}{(x^2+1)(x^2+2)} = \frac{Ax}{x^2+1} + \frac{Bx}{x^2+2}. $$
Multiplying both sides by $(x^2+1)(x^2+2)$ and dividing by $x$ (for $x\neq0$):
$$ 1 = A(x^2+2) + B(x^2+1). $$
Step 3 · Solve for the coefficients
Matching coefficients of $x^2$ and the constant term:
$$ A+B = 0, \qquad 2A+B = 1. $$
Subtracting the first from the second gives $A=1$, and so $B=-1$.
Step 4 · Integrate and combine
Substituting back:
$$ I = \int \left(\frac{x}{x^2+1} - \frac{x}{x^2+2}\right) dx. $$
Each term is a standard "numerator matches half the derivative of the denominator" pattern:
$$ I = \frac12\ln(x^2+1) - \frac12\ln(x^2+2) + C = \frac12\ln\left|\frac{x^2+1}{x^2+2}\right| + C. $$
Summary Table
| Coefficient | Value |
|---|---|
| $A$ | $1$ |
| $B$ | $-1$ |
| Antiderivative | $\tfrac12\ln\left|\tfrac{x^2+1}{x^2+2}\right|$ |
Additional Notes
- The integrand is defined for all real $x$, since $x^2+1>0$ and $x^2+2>0$ always.
- Because the numerator is odd (only $x$, no $x^3$ term), the partial fraction decomposition needs only $Ax/(x^2+1)$ and $Bx/(x^2+2)$ terms rather than the more general $Ax+B$ form used when both even and odd powers are present.
- The result can be checked by differentiating $\tfrac12\ln(x^2+1)-\tfrac12\ln(x^2+2)$ and confirming it returns $\dfrac{x}{(x^2+1)(x^2+2)}$.
Result
The solution to the integral is: