Problem 68 - Calculus for Biology and Medicine

Question

Evaluate each integral. $$ \int \frac{x^{2}+2 x}{(x+1)\left(x^{2}+2 x+2\right)} d x $$

Step-by-Step Solution

Answer

Decompose using partial fractions, solve for constants, integrate separately, and combine results.

Step 1: Identify the Integral Type

We need to evaluate the integral \(\int \frac{x^{2}+2 x}{(x+1)(x^{2}+2 x+2)} \, dx\). This rational function suggests the use of partial fraction decomposition since it involves a polynomial in the numerator and denominator.

Step 2: Apply Partial Fraction Decomposition

Decompose the integrand \(\frac{x^{2}+2x}{(x+1)(x^{2}+2x+2)}\) into simpler fractions. Assume it can be written as \(\frac{A}{x+1} + \frac{Bx+C}{x^{2}+2x+2}\). However, prior to this, ensure the partial fraction decomposition is set properly corresponding to the degrees of the polynomials.

Step 3: Equate and Solve for Constants

Write \(\frac{x^{2}+2x}{(x+1)(x^{2}+2x+2)} = \frac{A}{x+1} + \frac{Bx+C}{x^{2}+2x+2}\). Combine over a common denominator to form: \[x^2 + 2x = A(x^2 + 2x + 2) + (Bx+C)(x+1)\]. Expand and equate the coefficients for corresponding powers of \(x\). Solve for \(A, B,\) and \(C\).

Step 4: Evaluate Integrals of Simple Fractions

Once \(A, B,\) and \(C\) are found, integrate each fraction separately. \(\int \frac{A}{x+1} \, dx = A \ln|x+1| + C_1\). For the quadratic term \(x^2+2x+2\), use the substitution method for the \(Bx+C\) term if necessary, such as completing the square for the denominator.

Step 5: Solve the Integral for the Entire Expression

Combine the integrals to form the complete solution of the original integral. Make sure to include a constant of integration \(C\).

Step 6: Compute and Simplify as Necessary

Add the results of the separate integrals, and simplify if possible. Check against derivative rules to ensure your result is valid.

Key Concepts

Partial Fractions

Partial fraction decomposition is a method used to simplify complex rational expressions. It is extremely handy when integrating functions with polynomials in the numerator and denominator. In essence, it involves breaking down a complicated fraction into simpler, more manageable fractions. This process makes integration more straightforward.
In the given problem, the integrand \(\frac{x^{2}+2x}{(x+1)(x^{2}+2x+2)}\) needs to be expressed as a sum of simpler fractions. This is feasible because the degree of the polynomial in the numerator is lower than in the denominator.

• We assume a decomposition form like \(\frac{A}{x+1} + \frac{Bx+C}{x^{2}+2x+2}\).
• Equate the numerator of the original fraction to the expanded expression \(A(x^2 + 2x + 2) + (Bx+C)(x+1)\).
• Determine the constants \(A, B, \) and \(C\) through coefficient comparison of like terms.

Finding these constants can initially seem challenging, but with practice, it becomes much less daunting. The ability to spot opportunities for partial fractions is a powerful skill for anyone delving into integral calculus.

Integration Techniques

Once the original function is decomposed into partial fractions, the task shifts to integrating each separate fraction. Mastery of different integration techniques is crucial. Each situation might call for a specific approach, allowing us to utilize a range of methods effectively.
Basic integration skills are helpful, such as:

• Integration of simple rational functions like \(\frac{1}{x}\), which results in a natural logarithm.
• The power rule, which deals with polynomial terms.

For more complex terms like \(\frac{Bx+C}{x^2+2x+2}\), parts of the integration process may require additional steps like substitution or completing the square.
Completing the square in the denominator can sometimes transform the integral into a recognizable form, such as those leading to arctangent results. Practice with different techniques will build confidence in handling diverse integrals.

Polynomial Integration

Integrating polynomials is a fundamental skill in integral calculus. In our exercise, after applying partial fractions, we are left with simpler terms that often involve polynomial integration.
The power rule is the primary tool, which states that the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1}\) + C, applicable when \(neq -1\). This rule is straightforward and easily applied to polynomial terms.

• For simple fractions like \(\frac{A}{x+1}\), the result is an application of the natural logarithm function: \(A \ln|x+1| + C_1\).
• A term like \(\frac{Bx+C}{x^2+2x+2}\) may require an intermediary step, such as completing the square, before you can apply polynomial integration techniques effectively.

Polynomial integration simplifies to handling individual terms that fit the power rule format. The benefit of understanding polynomial integration is immense, as it stretches across broader calculus challenges.