Problem 23 - Calculus Early Transcendentals

Question

Integrals of \(\sin x\) and \(\cos x\) Evaluate the following integrals. $$\int \sin ^{2} x \cos ^{4} x d x$$

Step-by-Step Solution

Answer

The integral of \(\sin^2x\cos^4x\) with respect to x is: \(-\frac{9}{32}x + \frac{1}{8}\sin(2x) - \frac{15}{64}\sin(4x) + C\)

Step 1: Apply power reduction formulas

Rewrite the integral using the power reduction formulas: $$\int \sin^{2}x \cos^{4}x dx = \int \left(\frac{1-\cos(2x)}{2}\right) \left(\frac{1+\cos(2x)}{2}\right)^2 dx$$

Step 2: Simplify the expression

Now, simplify the expression inside the integral: $$\int \left(\frac{1-\cos(2x)}{2}\right) \left(\frac{1+\cos(2x)}{2}\right)^2 dx = \int \frac{(1-\cos(2x))(1+\cos(2x))^2}{16} dx$$

Step 3: Expand the expression

Expand the expression and distribute the terms: $$\int \frac{(1-\cos(2x))(1 + 2\cos(2x) + \cos^2(2x))}{16} dx = \int \frac{1 + 2\cos(2x) + \cos^2(2x) - \cos(2x) - 2\cos^2(2x) - \cos^3(2x)}{16} dx$$

Step 4: Simplify and apply the power reduction formula for \(\cos^2(2x)\)

Simplify the expression and apply the power reduction formula for \(\cos^2(2x)\): $$\int \frac{1 + \cos(2x) - 3\cos^2(2x) - \cos^3(2x)}{16} dx = \int \frac{1 + \cos(2x) - 3\frac{1+\cos(4x)}{2} - \frac{1+3\cos(4x)}{8}}{16} dx$$

Step 5: Simplify the expression

Simplify the expression inside the integral: $$\int \frac{1 + \cos(2x) - \frac{3 + 3\cos(4x)}{2} - \frac{1 + 3\cos(4x)}{8}}{16} dx = \int \frac{8 + 8\cos(2x) - 24 - 24\cos(4x) - 2 - 6\cos(4x)}{64} dx$$

Step 6: Simplify and integrate term by term

Simplify the expression and integrate each term: $$\int \frac{-18 + 8\cos(2x) - 30\cos(4x)}{64} dx = -\frac{18}{64} \int dx + \frac{8}{64} \int \cos(2x) dx - \frac{30}{64} \int \cos(4x) dx$$

Step 7: Integrate each term

Integrate each term to get the final answer: $$-\frac{18}{64}x + \frac{8}{64}\frac{\sin(2x)}{2} - \frac{30}{64}\frac{\sin(4x)}{4} + C = -\frac{9}{32}x + \frac{1}{8}\sin(2x) - \frac{15}{64}\sin(4x) + C$$

Key Concepts

Trigonometric Integrals

Trigonometric integrals involve the integration of products of trigonometric functions, such as \( \sin \) and \( \cos \). These integrals are common in calculus and often require special techniques or formulas to simplify. In the exercise, we are tasked with evaluating the integral of \( \sin^2 x \cos^4 x \). Such integrals often appear more manageable by transforming them using trigonometric identities or formulas.
The key to solving these integrals is recognizing patterns and applying appropriate techniques like power reduction, which convert complex products into simpler, more integrable forms. It's crucial to understand how the multiplication of trigonometric functions operates within the integral to begin the simplification process effectively.

• Firstly, identify the trigonometric functions involved and their powers.
• Look for recognizable patterns or identities that can simplify the expression.
• Apply specific approaches and formulas like power reduction when necessary to transform these products into sums or differences of simple functions.

Power Reduction Formula

The power reduction formula is a handy tool in integral calculus, particularly when dealing with products of trigonometric functions raised to a power, such as \( \sin^2 x \) or \( \cos^4 x \). These formulas help re-express such powers in terms of functions with lower powers, generally making the integration process much more straightforward.
For the integral \( \int \sin^2 x \cos^4 x dx \), the power reduction formulas allow us to break down \( \sin^2 x \) and \( \cos^4 x \) using identities like:

• \( \sin^2 x = \frac{1 - \cos(2x)}{2} \)
• \( \cos^2 x = \frac{1 + \cos(2x)}{2} \), and hence \( \cos^4 x = (\cos^2 x)^2 = \left(\frac{1 + \cos(2x)}{2}\right)^2 \)

These transformations simplify the integral into a manageable expression. This makes it easier to integrate by handling simpler functions rather than high power trigonometric polynomials. Practice using these formulas often will enhance a student's skill in tackling similar integrals with confidence.

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that hold for any value of the variable. These identities are particularly important in manipulating and simplifying complex trigonometric expressions.

In integral calculus, common identities used include:

• Pythagorean identities, e.g., \( \sin^2 x + \cos^2 x = 1 \)
• Angle sum and difference identities, e.g., \( \sin(2x) = 2\sin x \cos x \)
• Double angle identities, e.g., \( \cos(2x) = 2\cos^2 x - 1 \)

Such identities allow for the re-expression of integrands in a form that is simpler to integrate. For instance, in the given exercise, the expression is transformed using identities to facilitate easier integration. Understanding and remembering these identities aids students in recognizing the best approach to simplification and integration of trigonometric functions. Frequent use of these helpers makes complex calculus problems more approachable and more likely to yield a correct and swift solution.