Problem 15 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{1}{\sqrt{x} \sqrt{1-x}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\int \frac{1}{\sqrt{x} \sqrt{1-x}} dx\) is equal to \(-2 \sqrt{1-x} + C\).

Step 1: Identification

Identify that this integral is a candidate for substitution because it fits the form of \(\int \frac{1}{\sqrt{x} \sqrt{1-x}} dx\). A substitution with sine or cosine is usually helpful when dealing with square roots.

Step 2: Substitute

Let \(sin(\theta) = \sqrt{x}\), and by consequence, \(cos(\theta) = \sqrt{1-x}\). The differential dx can be obtained from this substitution. Take derivative of both sides to get \(cos(\theta) d\theta = \frac{1}{2\sqrt{x}} dx\). Now you can substitute the variables in the original integral.

Step 3: Simplify and Integrate

Substitute back into the integral and you have \(\int \frac{2 cos(\theta)}{sin(\theta) cos(\theta)} d\theta\) simplifying to \(\int \frac{2}{sin(\theta)} d\theta\). This can be integrated as \( -2 cos(\theta) + C \).

Step 4: Substitute Back

Substitute \(\theta\) back in terms of \(x\). Since you originally set \(sin(\theta) = \sqrt{x}\), it gives \(\theta = arcsin(\sqrt{x})\), and \( cos(\theta) = \sqrt{1-x}\). Therefore, \( -2 cos(\theta) + C = -2 \sqrt{1-x} + C\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in calculus, specifically in the process of integration. When you encounter an integral that seems complicated, substitution can often simplify the problem by introducing a new variable. This new variable is chosen to make integration more manageable.

For example, consider the integral \[\int \frac{1}{\sqrt{x} \sqrt{1-x}} dx \]Most will struggle with this form, that’s where substitution comes to the rescue. Look for a function within the integral whose derivative is also present, or can be produced by a simple manipulation. In this case, by letting \[sin(\theta) = \sqrt{x}\],we can express \[cos(\theta) d\theta\]as the differential of \[\sqrt{x}\], which is \[\frac{1}{2\sqrt{x}} dx\]. This transforms the integral into a trigonometric function which we are more comfortable dealing with.

To successfully use the substitution method, remember to:

• Choose the substitution that simplifies the integral.
• Express the differential dx in terms of the new variable.
• Perform the integration with respect to the new variable.
• Substitute back in terms of the original variable.

The method requires practice to identify the most effective substitution which often comes with experience.

Integration Techniques

Aside from substitution, there are other integration techniques that can be equally valuable when tackling different kinds of integral problems. Some of the most common methods include:

• Integration by Parts - useful when the integrand is the product of two functions.
• Partial Fraction Decomposition - effective for rational functions where the degree of the numerator is less than the degree of the denominator.
• Trigonometric Integration - for dealing with integrals involving trigonometric functions.
• Integration by Partial Fractions - useful when dealing with rational expressions.
• Improper Integrals – for integrals with infinite limits or integrands with asymptotes.

Each technique has specific scenarios where it applies best. Choosing the most efficient method to solve an integral can make the difference between a straightforward solution and a complex one. For complex integrals, a combination of methods might be necessary. It’s crucial to have a firm grasp on these various techniques, as they are vital tools in the arsenal of calculus.

Trigonometric Substitution

When the integrand involves square roots, particularly those following a Pythagorean identity, the trigonometric substitution method can be exceptionally useful. This technique borrows from the relationships inherent in trigonometry to simplify complex integrals.

Consider the previous example where we have square roots in the denominator. By observing that \[\sin^2(\theta) + \cos^2(\theta) = 1\],we can use this relation to substitute \[\sqrt{x}\]with \[\sin(\theta)\]and \[\sqrt{1-x}\]with \[\cos(\theta)\]. After this substitution, the integral becomes a standard trigonometric integral allowing easy integration.

Use trigonometric substitution when you see:

• An integrand that fits a trigonometric identity.
• A square root that can be expressed as a sine or a cosine function.
• An expression that would be simpler in a trigonometric form.

Making these connections between algebraic expressions and trigonometric identities can greatly simplify the process of integration, which is the heart of the trigonometric substitution method.