Problem 32 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: Setup the Integral Using Trigonometric Identity

The problem begins with the integral \( \int \sin^2 x \, dx \). Use the identity \( \sin^2 x = (\sin x)(\sin x) \) to express the integral.

Step 2: Apply Integration by Parts

To integrate \( \int \sin^2 x \, dx \) using integration by parts, let \( u = \sin x \) and \( dv = \sin x \, dx \). Then \( du = \cos x \, dx \) and \( v = -\cos x \). The integration by parts formula \( \int u \, dv = uv - \int v \, du \) gives:\[ \int \sin^2 x \, dx = (-\sin x \cos x) - \int (-\cos x)(\cos x) \, dx \]Which simplifies to:\[ = -\sin x \cos x + \int \cos^2 x \, dx \]

Step 3: Use the Pythagorean Identity

Apply the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to replace \( \cos^2 x \) in the integral. Thus, \( \cos^2 x = 1 - \sin^2 x \). Substitute this into the integral:\[ \int \cos^2 x \, dx = \int (1 - \sin^2 x) \, dx \]

Step 4: Simplify and Complete the Integration

Substitute back into the equation from Step 2:\[ \int \sin^2 x \, dx = -\sin x \cos x + \int (1 - \sin^2 x) \, dx \]This becomes:\[ = -\sin x \cos x + \int 1 \, dx - \int \sin^2 x \, dx \]Let \( I = \int \sin^2 x \, dx \). Thus:\[ I = -\sin x \cos x + x - I \]Which simplifies to:\[ 2I = -\sin x \cos x + x \]Finally, solving for \( I \):\[ I = \frac{1}{2} (-\sin x \cos x + x) + C \] where \( C \) is the integration constant.

Key Concepts

Trigonometric Identities

To tackle the problem of integrating \( \int \sin^2 x \, dx \), we must first understand the power of trigonometric identities. These are equations involving trigonometric functions that hold true for all values of the variable where the equations are defined. They help to simplify expressions and solve problems like calculus integrations efficiently.
One such identity is \( \sin^2 x = (\sin x)(\sin x) \). This expression allows breaking down powers of sine and helps in integration by parts later. Remember, using identities can significantly simplify the complexity of integrals and integrals involving multiple trigonometric terms.
Throughout trigonometry, identities can make or break the simplicity of solving equations, so knowing them becomes crucial as they can transform difficult integrals into simpler forms.

Integration by Parts

Integration by parts is an essential technique, especially for integrals involving products of different types of functions. The formula \( \int u \, dv = uv - \int v \, du \) derives from the product rule in differentiation and offers a method to integrate products by transferring the derivative from one function to another.
Applying this to \( \int \sin^2 x \, dx \), we make strategic choices for \( u \) and \( dv \). By letting \( u = \sin x \) and \( dv = \sin x \, dx \), we find that \( du = \cos x \, dx \) and \( v = -\cos x \). Substituting these into the integration by parts formula allows us to break down the problem, ultimately leading us to express the integral as:

• \( -\sin x \cos x + \int \cos^2 x \, dx \)

This formula is pivotal as it transforms the integral into a form that is simpler to handle or can be further simplified using identities.

Pythagorean Identity

The Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) is one of the most fundamental identities in trigonometry. It ties together the two primary trigonometric functions, sine and cosine, and is valid for every angle x. This identity provides a powerful tool to convert expressions between sine and cosine.
In the problem, we use this identity to manipulate the term \( \int \cos^2 x \, dx \). By expressing \( \cos^2 x \) as \( 1 - \sin^2 x \), the identity allows us to transform \( \int \cos^2 x \, dx \) into \( \int (1 - \sin^2 x) \, dx \).
This transformation and simplification underlines how identities can allow further integration progress. Ultimately, such identities are key for transforming and solving integrals that initially seem quite complex, turning them into something manageable or solvable with elementary principles.