Question

For the following exercises, find the indefinite integral. $$ \int \frac{d x}{1+x} $$

Short answer

The indefinite integral is \( \ln|1+x| + C \).

Step-by-step solution

Identify the Integral Form

The integral \( \int \frac{d x}{1+x} \) can be identified as the integral of a standard form \( \int \frac{1}{u} \, du \). In this case, \( u = 1 + x \), making it a logarithmic integration problem.

Determine the Substitution

To solve the integral, use the substitution \( u = 1 + x \), which implies \( du = dx \). This helps convert the integral in terms of \( u \).

Rewrite the Integral

Rewrite the integral \( \int \frac{d x}{1+x} \) as \( \int \frac{1}{u} \, du \). This step involves directly substituting \( u \) and \( du \) into the integral format.

Evaluate the Integral of \( \frac{1}{u} \, du \)

The integral \( \int \frac{1}{u} \, du \) is a standard result and evaluates to \( \ln|u| + C \), where \( C \) is the constant of integration.

Substitute Back the Original Variable

Substitute back the original expression for \( u \) to get the answer in terms of \( x \). Since \( u = 1 + x \), the integral becomes \( \ln|1+x| + C \).

Key concepts

Logarithmic Integration

In the world of calculus, logarithmic integration plays a crucial role when you encounter integrands of the form \( \int \frac{1}{x} \, dx \). This method involves recognizing and solving integrals that have a denominator closely resembling the expression \( 1+x \). A great example is the integral \( \int \frac{d x}{1+x} \), which is tackled using logarithmic integration. Due to the structure of the integrand, this type of problem leads to solutions involving natural logarithms.

• Identify the form: Look for integrals that resemble \( \int \frac{1}{x} \, dx \).
• Natural logarithm link: These integrals typically resolve into a natural logarithm, i.e., \( \ln|x| + C \).
• Common in many calculus problems, providing clarity and solving efficiency.

Substitution Method in Integration

The substitution method is a powerful technique used to simplify and solve integrals by changing variables. For the integral \( \int \frac{d x}{1+x} \), substitution simplifies the integration process by transforming it into an easier form to integrate. This involves changing the variable \( x \) into a new variable \( u \), such that the integral becomes manageable.

• Choose \( u \): Typically, you choose an expression that simplifies the integral. Here, \( u = 1 + x \).
• Find \( du \): Differentiate \( u \) to find \( du = dx \), ensuring that differentiation aligns with the transformation.
• Transform the integral: Substitute \( u \) and \( du \) into the original integral, converting it to \( \int \frac{1}{u} \, du \).

The key is to select substitutions that simplify the integration while reverting back to the original variable after integration.

Integration Techniques

Mastering integration involves understanding several techniques beyond basic integration rules. These techniques are useful when you face complex integrals, enhancing your problem-solving toolkit. This includes logarithmic integration and substitution, among others.Various techniques include:

• Logarithmic Integration: Used for integrals with a denominator that matches a simple linear form like \( 1+x \).
• Substitution Method: Change of variables to simplify the integral into a more familiar form.
• Integration by Parts: Useful for products of functions where integrated directly isn’t straightforward.
• Partial Fraction Decomposition: Simplifies integrands with rational expressions.

Combining these methods allows for tackling a wide range of integral problems effectively. A strong grasp of these techniques ensures that even the seemingly complex integrals become manageable and solvable.