Problem 28 - Biocalculus Calculus for the Life Sciences

Question

(a) Prove the reduction formula $$\int \cos ^{n} x d x=\frac{1}{n} \cos ^{n-1} x \sin x+\frac{n-1}{n} \int \cos ^{n-2} x d x$$ (b) Use part (a) to evaluate \(\int \cos ^{2} x d x\) (c) Use parts (a) and (b) to evaluate \(\int \cos ^{4} x d x\)

Step-by-Step Solution

Answer

The reduction formula is proven. \[ \int \cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + C \] \( \int \cos^4 x \, dx = \frac{1}{4} \cos^3 x \sin x + \frac{3}{8}(x + \sin x \cos x) + D \).

Step 1: Apply Integration by Parts

To derive the reduction formula \( \int \cos^n x \, dx \), use integration by parts. Let \( u = \cos^{n-1} x \) and \( dv = \cos x \, dx \). Thus, \( du = -(n-1) \cos^{n-2} x \sin x \, dx \) and \( v = \sin x \). Using the formula \( \int u \, dv = uv - \int v \, du \), we get \( \int \cos^n x \, dx = \cos^{n-1} x \sin x - \int \sin x (-(n-1) \cos^{n-2} x \sin x) \, dx \).

Step 2: Simplify the Resulting Expression

The integration by parts step leads to \( \int \cos^n x \, dx = \cos^{n-1} x \sin x + (n-1) \int \cos^{n-2} x \sin^2 x \, dx \). Use the identity \( \sin^2 x = 1 - \cos^2 x \) to substitute for \( \sin^2 x \). The integral now splits into two parts: \( (n-1) \int \cos^{n-2} x \, dx - (n-1) \int \cos^n x \, dx \).

Step 3: Solve the New Integral Equation

Rearrange to solve the integral equation: \( n \int \cos^n x \, dx = \cos^{n-1} x \sin x + (n-1) \int \cos^{n-2} x \, dx \). Divide through by \( n \) to isolate the original integral on one side, yielding the reduction formula: \( \int \cos^n x \, dx = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n} \int \cos^{n-2} x \, dx \).

Step 4: Apply the Reduction Formula for n=2

Substitute \( n = 2 \) into the reduction formula: \( \int \cos^2 x \, dx = \frac{1}{2} \cos x \sin x + \frac{1}{2} \int dx \). The integral of \( dx \) is \( x \), so the expression simplifies to \( \frac{1}{2} \cos x \sin x + \frac{1}{2} x + C \), where \( C \) is the integration constant.

Step 5: Simplify \( \int \cos^2 x \, dx \)

The expression for \( \int \cos^2 x \, dx \) using trigonometric identities is \( \frac{1}{2}(x + \sin x \cos x) + C \). Alternatively, this can also be expressed using \( \frac{1}{2}(x + \frac{1}{2} \sin(2x)) + C \).

Step 6: Apply Reduction Formula for n=4

Using the reduction formula for \( n = 4 \), substitute into the formula: \( \int \cos^4 x \, dx = \frac{1}{4} \cos^3 x \sin x + \frac{3}{4} \int \cos^2 x \, dx \). Substitute the expression found for \( \int \cos^2 x \, dx \) to evaluate this.

Step 7: Evaluate \( \int \cos^4 x \, dx \)

Using the result from Step 5, substitute \( \int \cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + C \) into the equation for \( \int \cos^4 x \, dx \). Simplifying gives \( \int \cos^4 x \, dx = \frac{1}{4} \cos^3 x \sin x + \frac{3}{8}(x + \sin x \cos x) + D \), where \( D \) is another constant of integration.

Key Concepts

Reduction Formula

The reduction formula is a recursive mathematical tool used to simplify complex integrals. In the context of trigonometric functions, it particularly comes in handy when dealing with powers of sine or cosine. The reduction formula for integrating powers of cosine, as derived in the exercise, is:\[ \int \cos^n x \, dx = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n-1}{n} \int \cos^{n-2} x \, dx\]This formula allows us to reduce the power of the cosine function involved in the integration process. By reducing the power step-by-step, we can gradually solve more complex integrals, like \( \int \cos^4 x \, dx \), by expressing them in terms of simpler integrals, such as \( \int \cos^2 x \, dx \). It's like peeling away layers of an onion, simplifying the problem progressively. Integration by parts and trigonometric identities often accompany this reduction to achieve the solution.

Trigonometric Identities

Trigonometric identities play a crucial role in solving integrals involving trigonometric functions. In the exercise provided, we use the identity:\[ \sin^2 x = 1 - \cos^2 x\]This identity allows us to transform the expression \( \int \cos^n x \, dx \) in step 2 into a more manageable form. By substituting \( \sin^2 x \) with \( 1 - \cos^2 x \), the integral becomes easier to split and solve.

• They help break down complex trigonometric functions into more familiar forms.
• You can derive identities to substitute values that facilitate integration.

Trigonometric identities are not just tools for algebraic simplification but are a powerful ally in integral calculations, allowing us to maneuver through seemingly difficult problems with elegance.

Definite Integrals

Definite integrals provide the area under the curve of a function between two specified limits. While the provided exercise focuses on indefinite integrals, understanding definite integrals gives broader context and application.When you solve \( \int \cos^2 x \, dx \) or \( \int \cos^4 x \, dx \), the fundamental result is an indefinite integral. However, if we integrate over a specific range, say from \( a \) to \( b \), the solution answers the question of the net area "between" the curve \( \cos^n x \) and the x-axis over that interval.

• Definite integrals measure real-world quantities, like areas or averages.
• Complements the theoretical understanding of indefinite integrals with practical applications.

Through understanding definite integrals, one can bridge abstract mathematical concepts with tangible outcomes, translating mathematical insight into meaningful real-world applications.