Problem 87 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int \frac{5 x^{2}}{x^{2}+1} d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( 5x - \arctan(x) + C \).

Step 1: Identify the Integration Technique

Look at the expression \( rac{5x^2}{x^2+1} \). Notice that the numerator can be split into two terms, as \( x^2 + (5x^2-x^2) = 5x^2 \). Use this to set up the integration.

Step 2: Rewrite the Integrand

Rewrite \( rac{5x^2}{x^2 + 1} \) as \( rac{x^2}{x^2 + 1} + rac{4x^2}{x^2 + 1} \). This simplifies to \( 1 + rac{4x^2}{x^2 + 1} \), because \( \frac{x^2}{x^2+1} = 1 - \frac{1}{x^2+1} \). Therefore, the integrand becomes \( 1 + 4 - \frac{1}{x^2 +1} \).

Step 3: Perform the Integration

Now the integral \( \int \left( 1 + 4 - \frac{1}{x^2 + 1} \right) \, dx \) can be separated as \( \int 1 \, dx + \int 4 \, dx - \int \frac{1}{x^2 + 1} \, dx \). These integrate to: - \( \int 1 \, dx = x \) - \( \int 4 \, dx = 4x \)- \( \int \frac{1}{x^2 + 1} \, dx = \arctan(x) \).

Step 4: Combine the Results

Combine the antiderivatives found in the previous step. The solution is \[ 4x + x - \arctan(x) + C \] where \( C \) is the constant of integration.

Key Concepts

Integration Techniques

Understanding different techniques in integration is crucial for computing complex indefinite integrals. In general, there are multiple methods, including substitution, integration by parts, and partial fraction decomposition. However, choosing the right method largely depends on the form of the integrand.
When you approach an integral, first analyze its components. Look for patterns or identities such as trigonometric functions or logarithmic forms that suggest a specific approach. In this exercise, we recognize the technique of dividing the expression into simpler parts as a suitable approach.

Rewrite the Integrand

Rewriting the integrand helps simplify the integration process. It involves breaking down or transforming the expression into more familiar or manageable components.
In the original problem, the expression is \( \frac{5 x^{2}}{x^{2}+1} \). By recognizing that \( x^2 + (5x^2-x^2) = 5x^2 \), you can express the integrand as \( \frac{x^2}{x^2 + 1} + \frac{4x^2}{x^2 + 1} \). This further simplifies to \( 1 + 4 \frac{x^2}{x^2 + 1} \) with \( \frac{x^2}{x^2 + 1} = 1 - \frac{1}{x^2+1} \).
By rewriting it, the problem becomes more approachable, separating it into parts where basic integration can be applied.

Perform the Integration

Once you've rewritten the integrand, it's time to perform the integration. This involves calculating the integral of each part separately.
The separated expression \( \int (1 + 4 - \frac{1}{x^2 + 1}) \, dx \) is integrated as follows:

• \( \int 1 \, dx = x \)
• \( \int 4 \, dx = 4x \)
• \( \int \frac{1}{x^2 + 1} \, dx = \arctan(x) \)

Each term is straightforward due to basic integration rules, such as integrating constants and recognizing standard antiderivatives.

Antiderivatives

Antiderivatives are functions whose derivative matches the given function. Finding them is the goal when performing indefinite integration.
In our exercise, after performing the integrals individually:

• The antiderivative of \( 1 \) is \( x \).
• The antiderivative of \( 4 \) is \( 4x \).
• The function \( \arctan(x) \) is a well-known antiderivative of \( \frac{1}{x^2 + 1} \).

Combining these results gives the complete antiderivative:
\[ 4x + x - \arctan(x) + C \], where \( C \) represents the constant of integration, necessary for indefinite integrals to express all possible antiderivatives.