Problem 88 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int \frac{2 x^{2}}{1+x^{2}} d x $$

Step-by-Step Solution

Answer

The integral of \( \int \frac{2x^2}{1+x^2} \, dx \) is \( x^2 - \ln|1+x^2| + C \).

Step 1: Identify the Type of Integral

Notice that the integrand is a rational function. A common technique for integrals of this form is to use substitution. Here, consider the substitution that might simplify the integrand.

Step 2: Choose Substitution

Let \( u = 1 + x^2 \). This gives us \( du = 2x \, dx \). Therefore, we can rearrange to find \( x \, dx = \frac{1}{2} du \). Also, rewrite \( 2x^2 = 2(x^2 + 1 - 1) = 2(x^2 + 1) - 2 \).

Step 3: Express the Integral in Terms of u

Substitute \( u = 1 + x^2 \) and \( x \, dx = \frac{1}{2} du \) into the integral to get:\[ \int \frac{2(x^2 + 1) - 2}{u} \frac{1}{2} du = \int \frac{x^2 + 1}{u} du - \int \frac{1}{u} \, du \]This simplifies to:\[ \int du - \int \frac{1}{u} \, du \]

Step 4: Solve Each Integral Separately

The integral \( \int du \) is simply \( u \). The integral \( \int \frac{1}{u} \, du \) is \( \ln|u| \). So the solution to the integral is:\[ u - \ln|u| + C \] where \( C \) is the constant of integration.

Step 5: Substitute Back in Terms of x

Since \( u = 1 + x^2 \), substitute back to get:\[ 1 + x^2 - \ln|1 + x^2| + C \]

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus to simplify integrals, particularly when dealing with composite functions. This method works by replacing a complex part of an integrand with a single variable (often denoted as \( u \)) which makes the integration process easier.

• Identify Part of the Integrand: First, locate a part of the integrand that could simplify the whole expression if substituted by \( u \). For example, in our original exercise, the natural choice is \( u = 1 + x^2 \).
• Determine \( du \): Once you've made the substitution, differentiate \( u \) to find \( du \). In our case, \( du = 2x \, dx \).
• Rewrite the Integral: Express the differential \( dx \) in terms of \( du \). Solve for \( x \, dx \) in terms of \( du \), like \( x \, dx = \frac{1}{2} du \).
• Simplify and Integrate: Substitute all expressions related to \( x \) with those involving \( u \) and simplify the integrand to make it easier to integrate.

Rational Functions

Rational functions are expressions which can be written as the quotient of two polynomials. These functions often appear in calculus problems, especially in integration challenges. The core challenge in integrating rational functions lies in simplifying complex fractions and finding suitable methods, such as substitution, to integrate them.

• Structure: Rational functions have the form \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.
• Decomposition: Sometimes, complex rational functions need to be broken into simpler fractions, especially when \( Q(x) \) has factors. However, in the given integral, a substitution alone was sufficient to simplify the expression for integration.
• Substitution Ability: Rewriting a rational function through substitution can transform a complicated rational expression into an integrable form. Like transforming \( \frac{2x^2}{1+x^2} \) into simpler terms by using \( u \) substitution in our exercise.

Integration Techniques

Integration techniques are varied strategies used to solve different integral problems. Not every problem can be solved using a single method, which is why having a variety of techniques at your disposal is critical.

• Basic Integration Rules: For simple polynomials, rational functions, and exponential functions, direct integration based on standard rules is often sufficient.
• Substitution Method: Often used when the integral involves a composite function or a substitution can simplify the equation, transforming it into a form that's easier to evaluate, as we've done with \( u = 1 + x^2 \).
• Partial Fractions: If the integrand is a rational function with irreducible factors, decomposing into partial fractions could be necessary. However, in our exercise, substitution was more straightforward.
• Natural Logarithms: Problems involving \( \frac{1}{x} \) or similar forms often lead to solutions involving natural logarithms, as seen where the solution includes \( \ln|u| \).

In practice, choosing which technique applies often depends on recognizing patterns in the integral, similar to identifying \( 1 + x^2 \) as a potential \( u \) substitution in a rational function.