Question

Use the double-angle formulas to evaluate the following integrals. \(\int_{0}^{\pi} \sin ^{4} x d x\)

Short answer

The integral evaluates to \(\frac{3}{8}\pi\).

Step-by-step solution

Use Identity for Sin^4

Recall the double-angle identity: \[\sin^2 x = \frac{1 - \cos 2x}{2}.\]First, use this to express \(\sin^4 x\) in terms of \(\cos 2x\):\[\sin^4 x = (\sin^2 x)^2 = \left(\frac{1 - \cos 2x}{2}\right)^2.\]

Simplify the Expression

Expand the expression for \(\sin^4 x\):\[\left(\frac{1 - \cos 2x}{2}\right)^2 = \frac{(1 - \cos 2x)^2}{4} = \frac{1 - 2\cos 2x + \cos^2 2x}{4}.\]

Use Double-Angle Formula for Cos^2

Substitute \(\cos^2 2x\) with another identity:\[\cos^2 2x = \frac{1 + \cos 4x}{2}. \]Thus, the expression becomes:\[\frac{1 - 2\cos 2x + \frac{1 + \cos 4x}{2}}{4} = \frac{1 - 2\cos 2x + \frac{1}{2} + \frac{\cos 4x}{2}}{4}.\]

Integrate the Simplified Expression

Combine all terms to create a single expression for integration:\[\frac{3}{8} - \frac{\cos 2x}{2} + \frac{\cos 4x}{8}.\]Now, integrate term by term:\[\int_{0}^{\pi} \left( \frac{3}{8} - \frac{\cos 2x}{2} + \frac{\cos 4x}{8} \right) dx.\]

Compute Each Integral

Calculate each integral separately:- \(\int \frac{3}{8} dx = \frac{3}{8}x + C\)- \(\int -\frac{\cos 2x}{2} dx = -\frac{1}{4}\sin 2x + C\)- \(\int \frac{\cos 4x}{8} dx = \frac{1}{32}\sin 4x + C\)Combine these results:\[\left[ \frac{3}{8}x - \frac{1}{4}\sin 2x + \frac{1}{32}\sin 4x \right]_{0}^{\pi}.\]

Evaluate the Definite Integral

Substitute the limits into the combined expression to find the value:\[\left( \frac{3}{8}\pi - \frac{1}{4}\sin(2\pi) + \frac{1}{32}\sin(4\pi) \right) - \left( 0 - 0 + 0 \right).\]Since \(\sin(2\pi) = 0\) and \(\sin(4\pi) = 0\), the final result is:\[\frac{3}{8}\pi.\]

Key concepts

Double-Angle Formulas

The double-angle formulas are essential tools in trigonometry, specifically for simplifying expressions and evaluating integrals. They allow us to express trigonometric functions with double angles, such as \(2x\) or \(4x\), in terms of single-angle functions, simplifying otherwise complex expressions.

One important double-angle formula used in this exercise is for \(\sin^2 x\), which is given by:

• \(\sin^2 x = \frac{1 - \cos 2x}{2}\)

This formula lets us rewrite \(\sin^4 x\) as \((\sin^2 x)^2\), eventually leading us to express \(\sin^4 x\) in terms of \(\cos 2x\). By simplifying the expression using the formula for \(\cos^2 2x\), we manage to turn the integral into more straightforward components.

Understanding how these double-angle identities work helps in decomposing higher power trigonometric expressions, which are often encountered in integrals. When integrals appear complex, using these identities provides a way to simplify the function into manageable parts that are easier to integrate.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. They serve as invaluable tools in calculus, particularly for transforming integrals into simpler forms.

Some key trigonometric identities involved in this exercise include:

• The identity for \(\sin^2 x\): \(\sin^2 x = \frac{1 - \cos 2x}{2}\)
• The identity for \(\cos^2 x\): \(\cos^2 x = \frac{1 + \cos 2x}{2}\)

These identities allow us to express powers of sine and cosine in forms that are better suited for integration. For instance, the use of \(\cos^2 2x = \frac{1 + \cos 4x}{2}\) in this exercise was crucial to break down \(\sin^4 x\) into simpler algebraic components.

Having a good grasp of these identities allows for creative manipulation of trigonometric expressions. This skill not only aids in solving integrals but also in proving equations, verifying complex solutions, and simplifying expressions across various mathematical fields.

Definite Integrals

Definite integrals represent the net area under a curve within a certain interval, and they are fundamental in calculus. The process involves evaluating the integral of a function over a specific range, marked by the integral limits.

In this exercise, the integral \(\int_{0}^{\pi} \sin^4 x \, dx\) requires evaluating the trigonometric expression within the interval from 0 to \(\pi\). After rewriting \(\sin^4 x\) using trigonometric identities and then simplifying it, you can integrate each part separately. Each resulting integral is evaluated using techniques learned from basic integration rules.

To compute a definite integral, follow these steps:

• Simplify the function using suitable identities or techniques.
• Integrate the resulting simpler expression term by term.
• Apply the limits to the antiderivative to find the exact area under the curve.

In this particular problem, the use of trigonometric identities and subsequent integration led to an end result—\(\frac{3}{8}\pi\)—after applying the limits of 0 to \(\pi\). Employing definite integrals in this way confirms the area occupied by the curve or the total accumulation of quantities represented by the function over the specified range.