Problem 31 - Calculus for Biology and Medicine

Question

Evaluating the integral \(\int \cos ^{2} x d x\) requires two steps. First, write $$ \cos ^{2} x=(\cos x)(\cos x) $$ and integrate by parts to show that $$ \int \cos ^{2} x d x=\sin x \cos x+\int \sin ^{2} x d x $$ Then, use \(\sin ^{2} x+\cos ^{2} x=1\) to replace \(\sin ^{2} x\) in the integral on the right-hand side, and complete the integration of \(\int \cos ^{2} x d x\).

Step-by-Step Solution

Answer

\( \int \cos^2 x \, dx = \frac{1}{2} (\cos x \sin x + x) + C \)

Step 1: Set up Integration by Parts

First, note that \( \cos^2 x = (\cos x)(\cos x) \). For integration by parts, recall the formula \( \int u \, dv = uv - \int v \, du \). Choose \( u = \cos x \) and \( dv = \cos x \, dx \). Then, \( du = -\sin x \, dx \) and \( v = \int \cos x \, dx = \sin x \).

Step 2: Apply Integration by Parts

Substitute the chosen functions into the integration by parts formula: \( \int \cos x \cos x \, dx = \cos x \sin x - \int \sin x (-\sin x) \, dx = \cos x \sin x + \int \sin^2 x \, dx \).

Step 3: Use Pythagorean Identity

Recall the identity \( \sin^2 x + \cos^2 x = 1 \). This gives us \( \sin^2 x = 1 - \cos^2 x \). Substitute \( \sin^2 x \) in the integral to obtain \( \int \sin^2 x \, dx = \int (1 - \cos^2 x) \, dx = \int 1 \, dx - \int \cos^2 x \, dx \).

Step 4: Simplify the Equation

From Step 2, we arrived at \( \int \cos^2 x \, dx = \cos x \sin x + \int \sin^2 x \, dx \). Substitute \( \int \sin^2 x \, dx = \int 1 \, dx - \int \cos^2 x \, dx \) from Step 3. Therefore, \( \int \cos^2 x \, dx = \cos x \sin x + \left( x - \int \cos^2 x \, dx \right) \).

Step 5: Solve for the Integral

Reorganize the equation from Step 4: \( \int \cos^2 x \, dx + \int \cos^2 x \, dx = \cos x \sin x + x \), which simplifies to \( 2 \int \cos^2 x \, dx = \cos x \sin x + x \). Thus, \( \int \cos^2 x \, dx = \frac{1}{2} (\cos x \sin x + x) + C \), where \( C \) is the constant of integration.

Key Concepts

Integration by Parts

Integration by parts is a handy method used in calculus to handle integrals where products are involved. The formula we use can be expressed as \( \int u \, dv = uv - \int v \, du \). It's like a magical tool helping us when standard integration methods can't do the trick easily.

To apply this method, you'll need to choose two components from the integral: \( u \) and \( dv \). Picking these parts correctly can be a bit of an art, requiring some practice to master. For instance, when integrating \( \int \cos^2 x \, dx \), we set \( u = \cos x \) and \( dv = \cos x \, dx \). Then, we compute \( du \) and \( v \):

• \( du = -\sin x \, dx \)
• \( v = \int \cos x \, dx = \sin x \)

By substituting these into the formula, the complex-looking integral transforms into something more manageable. We can then proceed with further simplification and integration.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental identities in trigonometry, expressing an ever-true relationship for sine and cosine. It tells us that \( \sin^2 x + \cos^2 x = 1 \) for any angle \( x \). This identity is particularly useful in diverse calculus problems, allowing for substitution and simplification.

In our calculus exercise, this identity becomes a crucial substitution tool. When we substitute \( \sin^2 x \) using the identity:

• \( \sin^2 x = 1 - \cos^2 x \)

We seamlessly transition from a challenging integral into one that is more straightforward. This step is essential because it reduces the complexity of the integral, making it easier to complete the evaluation. Keep this identity in your toolkit as it is often the key to solving trigonometric integrals.

Trigonometric Integrals

Trigonometric integrals involve functions like sine, cosine, and their powers. Evaluating these integrals is a frequent task in calculus. While they can seem intimidating at first, using identities and methods like integration by parts makes them manageable.

For the integral \( \int \cos^2 x \, dx \), we initially handle the problem by integrating by parts. Then, using the Pythagorean identity helps us further simplify the expression. Once in this simpler form, it is easier to integrate:

• Break down the power of the trigonometric function.
• Use identities to simplify and transform the expression.
• Apply integration methods to evaluate the integral component by component.

These techniques reduce the integral into well-known, easier-to-integrate parts, leading us to the solution efficiently. Developing familiarity with these steps deepens understanding and expands your problem-solving toolkit for calculus.