Q.Find: $\displaystyle\int \dfrac{x^{2}+x+1}{(x+2)(x^{2}+1)}\,dx$

The key idea is partial fraction decomposition for integrating rational functions.

1. Decompose the integrand into partial fractions:

Multiplying by the denominator $(x+2)(x^2+1)$ gives $x^{2}+x+1 = A(x^{2}+1) + (Bx+C)(x+2)$.

2. Solve for the constants $A, B, C$:

Setting $x=-2$, we get $3 = A(5) \implies A = \frac{3}{5}$.

Expanding and equating coefficients: $x^{2}+x+1 = (A+B)x^{2} + (2B+C)x + (A+2C)$.

From $A+B=1$, we have $\frac{3}{5}+B=1 \implies B=\frac{2}{5}$.

From $A+2C=1$, we have $\frac{3}{5}+2C=1 \implies 2C=\frac{2}{5} \implies C=\frac{1}{5}$.

Thus, the decomposition is $\dfrac{3}{5(x+2)} + \dfrac{2x+1}{5(x^{2}+1)}$.

3. Integrate each term: …

The problem asks us to find the integral of a rational function, which is a ratio of two polynomials. When the denominator of a rational function can be factored, and its degree is greater than the degree of the numerator, a powerful technique called partial fraction decomposition comes into play. This method allows us to break down a complex rational function into a sum of simpler rational functions, each of which is much easier to integrate using standard formulas.

The core idea is that if we have a fraction like $\frac{P(x)}{Q(x)}$, and $Q(x)$ can be factored into linear and/or irreducible quadratic factors, we can express $\frac{P(x)}{Q(x)}$ as a sum of terms. Each term corresponds to a factor in the denominator.

For a linear factor $(ax+b)$, the corresponding partial fraction term is $\frac{A}{ax+b}$.

For an irreducible quadratic factor $(ax^2+bx+c)$, the corresponding partial fraction term is $\frac{Bx+C}{ax^2+bx+c}$.

Once decomposed, we integrate each of these simpler terms.

Let's apply this method step-by-step.

1. Analyze the integrand and factor the denominator.

   The integrand is $\displaystyle\frac{x^{2}+x+1}{(x+2)(x^{2}+1)}$.

   First, we check the degrees of the numerator and denominator. The degree of the numerator ($x^2+x+1$) is 2. The degree of the denominator ($(x+2)(x^2+1) = x^3+2x^2+x+2$) is 3. Since the degree of the numerator is less than the degree of the denominator, we do not need to perform polynomial long division.

   The denominator is already factored into a linear term $(x+2)$ and an irreducible quadratic term $(x^2+1)$. The quadratic $x^2+1$ is irreducible over real numbers because its discriminant ($b^2-4ac = 0^2 - 4(1)(1) = -4$) is negative.

2. Set up the partial fraction decomposition.

   Based on the factors in the denominator, we can write the rational function as a sum of partial fractions:

Here, $A$, $B$, and $C$ are constants that we need to determine.

3. Solve for the constants $A$, $B$, and $C$.

To find these constants, we combine the terms on the right-hand side by finding a common denominator, which will be $(x+2)(x^2+1)$:

Now, we equate the numerators of the original expression and the combined partial fractions:

We can find the constants using a combination of strategic substitution and equating coefficients.

*   **Find $A$ using substitution:**
    To find $A$, we can choose a value of $x$ that makes the term $(Bx+C)(x+2)$ zero. This happens when $x+2=0$, i.e., $x=-2$.
    Substitute $x=-2$ into the equation:

*   **Find $B$ and $C$ using equating coefficients:**
    Now that we have $A$, we can substitute its value back into the equation and expand the right side:

    Group terms by powers of $x$:

    Now, we equate the coefficients of corresponding powers of $x$ on both sides:
    *   **Coefficient of $x^2$:**

    *   **Coefficient of $x$:**

        Substitute $B=\frac{2}{5}$:

    *   **Constant term:** (This serves as a check)

        Substitute $C=\frac{1}{5}$:

        The values are consistent. So, $A=\frac{3}{5}$, $B=\frac{2}{5}$, and $C=\frac{1}{5}$.

4. Rewrite the integral using the partial fractions.

Substitute the values of $A$, $B$, and $C$ back into the partial fraction decomposition:

Now, we can rewrite the original integral: …