Problem 31 - Calculus Single Variable

Question

Integrate by parts to evaluate the given definite integral. $$ \int_{0}^{1} 2 \arctan (x) d x $$

Step-by-Step Solution

Answer

\( \int_{0}^{1} 2 \arctan(x) \, dx = \frac{\pi}{2} - \ln(2) \).

Step 1: Identify Parts for Integration by Parts

We need to use integration by parts, where the formula is \( \int u \, dv = uv - \int v \, du \). Let \( u = \arctan(x) \) and \( dv = 2 \, dx \). Then, \( du = \frac{1}{1+x^2} \, dx \) and integrating \( dv \) gives \( v = 2x \).

Step 2: Apply the Integration by Parts Formula

Substituting into the integration by parts formula, we have: \[ \int_{0}^{1} 2 \arctan(x) \, dx = \left[ 2x \arctan(x) \right]_{0}^{1} - \int_{0}^{1} 2x \cdot \frac{1}{1+x^2} \, dx \].

Step 3: Evaluate the First Term

Calculate \( \left[ 2x \arctan(x) \right]_{0}^{1} \): - At \( x = 1 \), we get \( 2 \cdot 1 \cdot \frac{\pi}{4} = \frac{\pi}{2} \). - At \( x = 0 \), we get \( 2 \cdot 0 \cdot \arctan(0) = 0 \). So, \( \left[ 2x \arctan(x) \right]_{0}^{1} = \frac{\pi}{2} \).

Step 4: Evaluate the Second Integral

To solve \( \int_{0}^{1} \frac{2x}{1+x^2} \, dx \), use substitution: let \( z = 1 + x^2 \), then \( dz = 2x \, dx \).- Change of limits: when \( x = 0 \), \( z = 1 \) and when \( x = 1 \), \( z = 2 \).- The integral becomes \( \int_{1}^{2} \frac{1}{z} \, dz \), which evaluates to \( \ln(z) \big|_{1}^{2} = \ln(2) - \ln(1) = \ln(2) \).

Step 5: Combine Results

Substitute back into the formula from Step 2:\[ \int_{0}^{1} 2 \arctan(x) \, dx = \frac{\pi}{2} - \ln(2) \].

Key Concepts

Definite Integrals

Definite integrals are a fundamental part of calculus, essential for calculating the area under a curve between two points. In our exercise, we have a specific range from 0 to 1. Unlike indefinite integrals, which result in a family of functions, definite integrals provide a numerical result. The process involves finding an antiderivative, or a integral, of the function, then evaluating it at the upper and lower limits of integration.
To compute a definite integral:

• Find the antiderivative of the function.
• Evaluate this antiderivative at the upper limit and at the lower limit.
• Subtract the two results to find the area under the curve.

This is crucial for understanding areas, volumes, and even in applying the fundamental theorem of calculus, which links the concepts of differentiation and integration together. In our particular exercise, the definite integral gives us a precise value for the integration between the boundaries provided.

Integration with Substitution

Integration with substitution is a powerful technique used to simplify complex integrals. This method is similar to the reverse of the chain rule for differentiation. It's especially helpful when dealing with integrals that involve composite functions or when direct integration becomes difficult.
In substitution, you aim to replace a complicated expression with a single variable which makes the integration easier to handle.

• Identify the part of the function to substitute, often a composite or inner function.
• Make the substitution by setting it equal to a new variable.
• Find a differential for the substitution variable to replace the corresponding part in the original integral.
• Change the limits of integration if evaluating a definite integral.

In our example, we used substitution to solve \( \int \frac{2x}{1+x^2} \, dx \) by letting \( z = 1 + x^2 \). This substitution simplifies the integral considerably, transforming it into a logarithmic form that is straightforward to evaluate.

Arctan Integration

Arctan, or the inverse tangent function, appears in many integrals, often requiring specialized techniques for effective integration due to its distinct behavior and periodic nature. Integrating functions involving arctan can be challenging due to its derivative \( \frac{1}{1+x^2} \), which doesn't align perfectly with standard power or exponential integrals.
When integrating products involving arctan, like in our original problem, integration by parts emerges as a fitting method.

• Integration by parts lets us decompose the integral of a product of functions into simpler parts.
• This technique is patterned after the product rule for derivatives and can untangle complex integrations involving arctan.
• Using arctan functions often means paying attention to boundaries and considering potential simplifications, especially when dealing with definite integrals.

By identifying \( u = \arctan(x) \) and \( dv = 2 \, dx \), we parsed the integral into manageable components, eventually arriving at the solution through systematic application of the integration by parts formula.