Problem 35 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the indefinite integral. \(\int \frac{1+x}{1+x^{2}} d x\)

Step-by-Step Solution

Answer

The integral evaluates to \( \arctan(x) + \frac{1}{2} \ln|1+x^2| + C \).

Step 1: Recognize the Fraction Structure

The integrand \( \frac{1+x}{1+x^2} \) can be split into two separate terms: \( \frac{1}{1+x^2} \) and \( \frac{x}{1+x^2} \). This separation will help us evaluate the integral.

Step 2: Integrate the First Term

The first part of the integral is \( \int \frac{1}{1+x^2} \, dx \). This is a standard integral that equals \( \arctan(x) + C_1 \).

Step 3: Integrate the Second Term

The second part is \( \int \frac{x}{1+x^2} \, dx \). Notice that the derivative of \(1+x^2\) is \(2x\), so this part can be simplified using a substitution. Let \( u = 1+x^2 \), then \( du = 2x \, dx \), or \( \frac{1}{2} du = x \, dx \). The integral becomes \( \frac{1}{2} \int \frac{1}{u} \, du = \frac{1}{2} \ln|u| + C_2 \). Substituting back, we get \( \frac{1}{2} \ln|1+x^2| + C_2 \).

Step 4: Combine the Results

Now, combine the results of the two integrals: \( \arctan(x) + \frac{1}{2} \ln|1+x^2| + C \), where \( C \) is the constant of integration that includes both \( C_1 \) and \( C_2 \).

Key Concepts

Integration Techniques

Integration techniques are crucial tools in calculus and allow us to find the antiderivative of a function, which in turn helps us understand the area under a curve. One of the basic techniques is recognizing patterns in the integrand that simplify the process.

• If parts of the integrand remind you of derivatives or standard integrals, this might allow you to break down the integral into simpler parts.
• Standard integrals, such as \( \int \frac{1}{1+x^2} \, dx \), should be easily recognizable. This particular integral results in \( \arctan(x) \), which is a common form.
• Splitting complex fractions into simpler terms can also be beneficial as seen in this problem, where we separated \( \frac{1+x}{1+x^2} \) into two terms to deal with them individually.

Exploring different techniques makes tackling integrals less daunting and more systematic.

Trigonometric Integrals

Trigonometric integrals often appear in calculus, not just in trigonometry problems. They involve trigonometric functions and can be solved using patterns and substitutions. In this exercise, our task was made easier by recognizing the form \( \frac{1}{1+x^2} \). This is a standard integral related to trigonometric functions.
The integral of \( \frac{1}{1+x^2} \) is \( \arctan(x) + C_1 \). Why? Because the derivative of \( \arctan(x) \) is precisely \( \frac{1}{1+x^2} \). This relationship is particularly useful and appears frequently, so it's beneficial to remember this form.
Whenever you see fractions that resemble inverse trigonometric functions, check if they match these standard forms. Doing so can significantly speed up finding solutions.

Substitution Method

The substitution method is a powerful integration technique used to simplify complicated integrand expressions. Think of it as reversing the chain rule for derivatives. In this method, we replace a part of the integrand with a new variable to make the integration process simpler.
For the given integral \( \int \frac{x}{1+x^2} \, dx \), notice that the denominator’s derivative, \( 2x \), closely resembles the numerator. This makes a substitution promising.

• Set \( u = 1 + x^2 \). This makes \( du = 2x \, dx \), leading to \( \frac{1}{2} du = x \, dx \).
• The integral then becomes \( \frac{1}{2} \int \frac{1}{u} \, du \), which is a simpler form to integrate.
• This transforms the original expression into a natural logarithm: \( \frac{1}{2} \ln|u| + C_2 \).

Substitution simplifies integration by reducing problems into familiar forms, making them easier to solve. As a result, understanding and mastering this technique is essential for success in calculus.