Evaluate the integral: I = integral (x² + 2)/(x² + 1) dx
Evaluate the integral: $$ I = \int \frac{x^2 + 2}{x^2 + 1} \, dx. $$
Explanation of Method
The same "match the denominator" strategy used for improper rational fractions applies here: since the numerator's degree equals the denominator's degree, split off a copy of the denominator from the numerator so that only a constant remainder is left over the denominator. What remains is the standard arctangent integral.
- Rewrite the numerator so it contains a full copy of the denominator.
- Split the fraction into a constant term and a proper fraction.
- Integrate each piece, using the standard form for $\int \frac{dx}{x^2+1}$.
- Combine the results and add the constant of integration.
Step 1 · Split the fraction
Rewrite the integrand as:
$$ \frac{x^2+2}{x^2+1} = \frac{x^2+1}{x^2+1} + \frac{1}{x^2+1}. $$
Simplify the terms:
$$ \frac{x^2+2}{x^2+1} = 1 + \frac{1}{x^2+1}. $$
Thus, the integral becomes:
$$ I = \int \left(1 + \frac{1}{x^2+1}\right) dx. $$
Step 2 · Split the integral
Split the integral into two simpler integrals:
$$ I = \int 1\,dx + \int \frac{1}{x^2+1}\,dx. $$
Step 3 · Evaluate each integral
The first integral is straightforward:
$$ \int 1\,dx = x. $$
The second integral, $\displaystyle \int \frac{1}{x^2+1}\,dx$, is a standard integral:
$$ \int \frac{1}{x^2+1}\,dx = \arctan(x). $$
Step 4 · Combine the results
Combine the results of the two integrals:
$$ I = x + \arctan(x) + C, $$
where $C$ is the constant of integration.
Summary Table
| Term | Type | Antiderivative |
|---|---|---|
| $1$ | Constant term | $x$ |
| $1/(x^2+1)$ | Standard arctangent form | $\arctan(x)$ |
Additional Notes
- The integrand is defined for all real $x$, since $x^2+1 \neq 0$ always.
- Compare with Problem 9: there the leftover constant was subtracted ($x^2=(1+x^2)-1$), while here it is added ($x^2+2=(x^2+1)+1$) — the sign of the remaining term determines whether the arctangent is added or subtracted in the final answer.
- The result can be checked by differentiating $x + \arctan(x)$ and confirming it returns the original integrand.
Result
The solution to the integral is: