Problem 15 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-34\) without using tables. $$ \int_{0}^{1} \frac{\theta+1}{\sqrt{\theta^{2}+2 \theta}} d \theta $$

Step-by-Step Solution

Answer

The integral evaluates to \( \sqrt{3} \).

Step 1: Simplify the Integral Expression

The integral given is \( \int_{0}^{1} \frac{\theta+1}{\sqrt{\theta^{2}+2\theta}} \, d\theta \). We need to simplify the expression inside the square root first. Notice that \( \theta^2 + 2\theta \) can be rewritten as \( (\theta+1)^2 - 1 \). So, the integral becomes \( \int_{0}^{1} \frac{\theta+1}{\sqrt{(\theta+1)^2 - 1}} \, d\theta \).

Step 2: Use Substitution

To simplify further, use the substitution \( u = \theta + 1 \). This implies \( du = d\theta \) and changes the limits of integration from \( \theta = 0 \) to \( u = 1 \), and from \( \theta = 1 \) to \( u = 2 \). Thus, the integral becomes \( \int_{1}^{2} \frac{u}{\sqrt{u^2 - 1}} \, du \).

Step 3: Identify the Integral Form

The integral \( \int \frac{u}{\sqrt{u^2 - 1}} \, du \) is a standard integral form. Recognize this as the derivative of \( \sqrt{u^2 - 1} \). The integral of the derivative will give us \( \sqrt{u^2 - 1} \).

Step 4: Evaluate the Integral

Integrate to get \( \sqrt{u^2 - 1} \). Evaluating this from 1 to 2 gives \( \sqrt{2^2 - 1} - \sqrt{1^2 - 1} \), which is \( \sqrt{3} - 0 = \sqrt{3} \).

Step 5: Write the Final Answer

Thus, the value of the integral \( \int_{0}^{1} \frac{\theta+1}{\sqrt{\theta^{2}+2\theta}} \, d\theta \) is \( \sqrt{3} \).

Key Concepts

Integral Substitution

One of the powerful tools in calculus and particularly in integration is substitution. This technique essentially turns a complicated integral into a much simpler one by changing the variable. In the exercise, we used the substitution \( u = \theta + 1 \). This allowed us to convert the integral into a form that is more straightforward to handle. When performing substitution:

• Choose a substitution that simplifies the integral, such as transforming a difficult expression into a standard one.
• Calculate the differential, which in this case was \( du = d\theta \).
• Change the limits of integration according to the substitution. Here, \( \theta = 0 \) transforms to \( u = 1 \) and \( \theta = 1 \) to \( u = 2 \).
• Substitute back, evaluate, and simplify.

This process transforms integrals that seem complex or unsolvable at first glance into problems that can often be solved using basic integral formulas.

Definite Integral

Definite integrals are used to calculate the exact area under a curve between two points, which in this exercise are \( 0 \) and \( 1 \). After performing the substitution, the limits of integration were updated from \( \theta \) to \( u \). The definition of a definite integral emphasizes evaluating the antiderivative at the upper and lower bounds and using these to find the result.

• Once the integral is solved, substitute the limits back into the resulting expression.
• This yields a numerical value which, for our exercise, was \( \sqrt{3} \).

Finding the definite integral involves ensuring that you properly convert and adapt these limits when performing substitution, so that the numerical evaluation is an accurate representation of the original problem's solution.

Integration Techniques

The exercise demonstrates several techniques that are commonly used in calculus to solve integrals. These techniques help break down complex integrals into solvable parts:

• Simplification: Recognizing patterns and relationships that allow for simplification, such as observing \( \theta^2 + 2\theta = (\theta+1)^2 - 1 \).
• Substitution: Using a substitution like \( u = \theta + 1 \) to shift the integral to a more familiar form.
• Recognition of Standard Forms: Identifying standard integral forms, such as recognizing \( \frac{u}{\sqrt{u^2 - 1}} \) as the derivative of \( \sqrt{u^2 - 1} \).

Each integration technique builds on identifying the nature of the integrand and using the best strategy to simplify or reshape it into something more workable. This skill becomes a valuable toolset for tackling a wide variety of integral problems.

Mathematical Simplification

Simplification is often the first step in addressing complex integrals. By rewriting expressions in a clearer form, we can often reveal a path towards solving an integral. In this exercise, simplifying \( \theta^2 + 2\theta \) to \( (\theta+1)^2 - 1 \) was crucial. Simplification strategies include:

• Factoring: Where applicable, to rewrite expressions in easier forms.
• Recognizing Common Patterns: Such as perfect squares, which can simplify square roots or other complex expressions.
• Utilizing Identities: Making use of algebraic identities can simplify trigonometric, logarithmic, or polynomial integrands.

The simplification process not only clarifies the integral at hand but also aids in decision-making for subsequent steps like substitution or applying specific integral forms. It's pivotal in developing a strategy that brings the integral into a solvable format.