Problem 26 - Biocalculus Calculus for the Life Sciences

Question

Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. \(\int \sin ^{4} x d x\)

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{3x}{8} - \frac{1}{4}\sin(2x) + \frac{1}{32}\sin(4x) + C\), same as using tables.

Step 1: Use Trigonometric Identity

First, we can simplify \(\sin^4 x\) using trigonometric identities. Recall that \(\sin^2 x = \frac{1 - \cos(2x)}{2}\). Therefore, \(\sin^4 x = (\sin^2 x)^2 = \left(\frac{1 - \cos(2x)}{2}\right)^2\).

Step 2: Expand the Expression

Expand the expression: \(\left(\frac{1 - \cos(2x)}{2}\right)^2 = \frac{1 - 2\cos(2x) + \cos^2(2x)}{4}\). Use the identity \(\cos^2(2x) = \frac{1 + \cos(4x)}{2}\) to further simplify: \(\cos^2(2x) = \frac{1 + \cos(4x)}{2}\).

Step 3: Substitute and Simplify

Substitute back into the equation: \(\sin^4 x = \frac{1 - 2\cos(2x) + \frac{1 + \cos(4x)}{2}}{4} = \frac{3}{8} - \frac{\cos(2x)}{2} + \frac{\cos(4x)}{8}\).

Step 4: Integrate Term by Term

Integrate each term individually:1. \(\int \frac{3}{8} \, dx = \frac{3x}{8}\)2. \(\int -\frac{\cos(2x)}{2} \, dx = -\frac{1}{4}\sin(2x)\)3. \(\int \frac{\cos(4x)}{8} \, dx = \frac{1}{32}\sin(4x)\)Thus, the integral \(\int \sin^4 x \, dx = \frac{3x}{8} - \frac{1}{4}\sin(2x) + \frac{1}{32}\sin(4x) + C\).

Step 5: Compare With Tables

Comparing our result to a table of integrals, we find that they often use another form such as \(\int \sin^4 x \, dx = \frac{3}{8}x - \frac{1}{4}\sin(2x) + \frac{1}{32}\sin(4x) + C\), which is equivalent due to elementary trigonometric identities or slightly different manipulations.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in integral calculus. They help simplify complex expressions and make them more manageable for integration. In this exercise, we focus on the trigonometric function \(\sin^4 x\). We can use the identity \(\sin^2 x = \frac{1 - \cos(2x)}{2}\) to simplify the problem.

• The identity \(\sin^2 x\) transforms higher powers of sine into expressions involving cosine, making integration simpler.
• By using \(\sin^4 x = (\sin^2 x)^2\), this becomes \((\frac{1 - \cos(2x)}{2})^2\).

Expanding and applying identities like \(\cos^2(2x) = \frac{1 + \cos(4x)}{2}\) further simplifies the expression. These steps reduce the complexity and prepare the terms for straightforward integration.

Computer Algebra Systems

Computer algebra systems (CAS) are powerful tools in calculus that help evaluate integrals quickly and accurately. In this exercise, using a CAS can save time and offer precise solutions.

• CAS can automatically apply trigonometric identities and perform algebraic manipulation to simplify integrands like \(\sin^4 x\).
• They promptly calculate definite and indefinite integrals, showing results in exact form.

For students, a CAS serves as a helpful aid to verify their manual calculations. It not only supports checking work but also deepens understanding by displaying step-by-step solutions. This can reinforce concepts and teach valuable problem-solving techniques through repetition and exploration.

Step-by-Step Solutions

Step-by-step solutions are pivotal in learning calculus as they guide through the logical process of solving integrals. This method involves breaking down each task to make the overall integration more approachable.
In our specific exercise:

• We first rewrite \(\sin^4 x\) using trigonometric identities to simplify the expression.
• Next, expand and substitute to clarify the terms needing integration.
• Finally, integrate term-by-term instead of in one bulk, making the operation simpler.

Steps like these, with each component addressed logically, help prevent errors that might occur when directly jumping into complex integral calculations. By encouraging a systematic approach, students build confidence and competence in tackling integrals both manually and digitally. This method helps ensure you understand the transformations happening in each step, from using identities to algebraic simplifications.