Problem 12 - Calculus

Question

Find the indefinite integrels in \(1-20\). \(\int t \sqrt{1-t^{2}} d t\)

Step-by-Step Solution

Answer

The indefinite integral is \( -\frac{1}{3} (1-t^{2})^{3/2} + C \).

Step 1: Identify the Integral

We want to solve the integral \( \int t \sqrt{1-t^2} \, dt \). This is a type of trigonometric substitution problem.

Step 2: Choose a Trigonometric Substitution

Let \( t = \sin \theta \). Therefore, \( dt = \cos \theta \, d\theta \) and \( \sqrt{1-t^2} = \sqrt{1-\sin^2\theta} = \cos\theta \).

Step 3: Rewrite the Integral

Substitute \( t = \sin \theta \) and \( \sqrt{1-t^2} = \cos \theta \) along with \( dt = \cos \theta \, d\theta \) into the integral to get: \( \int \sin \theta \cos \theta \cdot \cos \theta \, d\theta = \int \sin \theta \cos^2 \theta \, d\theta \).

Step 4: Simplify the Integral

Using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \), the integral becomes: \( \int \sin \theta (1 - \sin^2 \theta) \, d\theta = \int \sin \theta \, d\theta - \int \sin^3 \theta \, d\theta \).

Step 5: Integrate Using Reduction Formula

For \( \int \sin^3 \theta \, d\theta \), use \( \int \sin^n \theta \, d\theta = -\frac{1}{n} \sin^{n-1} \theta \cos \theta + \frac{n-1}{n} \int \sin^{n-2} \theta \, d\theta \). Applying this gives: \( \int \sin^3 \theta \, d\theta = -\frac{1}{3} \sin^2 \theta \cos\theta + \frac{2}{3} \int \sin\theta \, d\theta \).

Step 6: Evaluate the Simpler Integrals

Compute \( \int \sin \theta \, d\theta = -\cos\theta + C_1 \) and solving \( \int \sin^3 \theta \, d\theta = -\frac{1}{3} \sin^2 \theta \cos\theta - \frac{2}{3} \cos\theta + C_2 \).

Step 7: Substitute Back After Simplification

After evaluating \( \int \sin \theta \, d\theta \) and completing \( \int \sin^3 \theta \, d\theta \), substitute back \( \sin \theta = t \) and \( \cos \theta = \sqrt{1 - t^2} \) to get the final result: \( \int t \sqrt{1-t^2} \, dt = \frac{1}{3} (1-t^2)^{3/2} + C \).

Step 8: Write the Final Answer

Thus, the indefinite integral is \( \int t \sqrt{1-t^{2}} \, dt = -\frac{1}{3} (1-t^{2})^{3/2} + C \).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique used to solve integrals involving square roots, especially when variables are expressed as the sum or difference of squares. This method simplifies integration by substituting a trigonometric function for the variable. For example, when dealing with an integral like \( \int t \sqrt{1-t^2} \, dt \), the substitution \( t = \sin \theta \) simplifies the integral significantly.

• Set \( t = \sin \theta \), leading to \( dt = \cos \theta \, d\theta \).
• The expression \( \sqrt{1-t^2} \) transforms into \( \cos \theta \), thanks to the identity \( 1-\sin^2 \theta = \cos^2 \theta \).

This approach reduces the difficulty of the integral by turning complex algebraic expressions into much simpler trigonometric ones. As a result, the problem becomes a trigonometric integral which is easier to manage.

Reduction Formula

The reduction formula is a systematic way to break down complex integrals involving powers of trigonometric functions into simpler ones. It is particularly useful when direct integration isn't straightforward. In our example, after simplifying with trigonometric substitution, we face \( \int \sin^3 \theta \, d\theta \). The reduction formula for sine is:\[ \int \sin^n \theta \, d\theta = -\frac{1}{n} \sin^{n-1} \theta \cos \theta + \frac{n-1}{n} \int \sin^{n-2} \theta \, d\theta \]

• This turns \( \int \sin^3 \theta \, d\theta \) into a combination of simpler integrals, making it easier to solve step by step.
• Here, \( n = 3 \) simplifies to working with \( \sin \theta \) and \( \sin^2 \theta \), which are much easier to integrate.

Using a reduction formula not only simplifies the process but helps manage any power of sine or cosine efficiently, ensuring the solution remains accurate.

Integration Techniques

Integration techniques encompass a variety of strategies designed to simplify the computation of integrals. When tasks like \( \int t \sqrt{1-t^2} \, dt \) arise, choosing the proper technique is crucial. In this context, combining different techniques like trigonometric substitution and reduction formulas builds a cohesive strategy.Here are a few essential integration techniques to consider:

• Substitution, which can transform a challenging integral into an easier one by changing variables.
• Partial fraction decomposition, useful for fractions with polynomials in the denominator.
• Integration by parts, helpful for integrals of products of functions.

Approaching an integral with these techniques allows for flexibility and a better understanding of the structure of mathematical expressions. Learning to identify which method suits a particular problem is key to mastering integration.