Problem 46 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int \sin ^{6}(5 \theta) \cos (5 \theta) d \theta$$

Step-by-Step Solution

Answer

\( \int \sin^6(5\theta) \cos(5\theta) d\theta = \frac{1}{35} \sin^7(5\theta) + C \).

Step 1: Substitution

Let us use the substitution method to simplify the integral. Start by letting \( u = \sin(5\theta) \). Consequently, the derivative \( \frac{du}{d\theta} = 5\cos(5\theta)\). Rewriting, we get \( du = 5\cos(5\theta)d\theta \), so \( \cos(5\theta)d\theta = \frac{du}{5} \). Now substitute in the integral: \[ \int \sin^6(5\theta) \cos(5\theta) d\theta = \int u^6 \cdot \frac{du}{5} \].

Step 2: Integrate with respect to \(u\)

We have transformed the integral into \( \frac{1}{5}\int u^6 du \). Find the integral with respect to \( u \): \[ \frac{1}{5} \left( \frac{u^7}{7} \right) \].

Step 3: Substitute back for \( \theta \)

Substitute \( u = \sin(5\theta) \) back into the equation to express the integral in terms of \( \theta \): \[ \frac{1}{5} \cdot \frac{1}{7} \sin^7(5\theta) = \frac{1}{35} \sin^7(5\theta) \]. Therefore, the integral is \( \frac{1}{35} \sin^7(5\theta) + C \), where \( C \) is the constant of integration.

Step 4: Verification by Differentiation

Differentiate the result to verify it matches the original integrand. The derivative of \( \frac{1}{35} \sin^7(5\theta) + C \) with respect to \( \theta \) is \( \frac{7}{35} \cdot 5 \cdot \sin^6(5\theta) \cdot \cos(5\theta) \). Simplifying, we get \( \sin^6(5\theta) \cos(5\theta) \), confirming that our indefinite integral is correct.

Key Concepts

Indefinite Integral

An indefinite integral is a type of integral that does not have specified limits. It represents a family of functions and is often denoted with the integral sign without upper and lower limits. The goal is to find a function whose derivative matches the given integrand. This is crucial for understanding how functions accumulate area.

Indefinite integrals result in a general form plus an arbitrary constant. This constant, usually represented by \( C \), accounts for all possible vertical shifts of the antiderivative.

• The general formula for an indefinite integral of a function \( f(x) \) is \( \int f(x) \, dx = F(x) + C \) where \( F(x) \) is the antiderivative of \( f(x) \).
• The constant of integration \( C \) is important because indefinite integrals represent families of functions.

Recognizing and understanding indefinite integrals is fundamental in calculus and helps in solving various types of differential equations and in the evaluation of definite integrals.

Check by Differentiation

Once an indefinite integral has been calculated, it is good practice to verify the result through differentiation. This involves taking the derivative of the antiderivative function we obtained and checking if it matches the original integrand.

The process ensures that no mistakes were made during manipulation and transformation steps in finding the integral.

• To check the result, differentiate the function \( F(x) + C \).
• If doing so gives back the original integrand \( f(x) \), then the integration process was correct.

In the given problem, the integration result \( \frac{1}{35} \sin^7(5\theta) + C \) was differentiated, yielding the original integrand \( \sin^6(5\theta) \cos(5\theta) \). This confirmed our solution was correct.

Substitution Method

The substitution method is a technique used to simplify the process of finding an integral. It involves changing the variable of integration to make the integral easier to solve. By replacing parts of the integral with a new variable, difficult integrals can often be turned into simpler ones.

This method is particularly useful when dealing with integrals involving compositions of functions.

• Identify a substitution \( u = g(\theta) \), which will simplify the given integral.
• Calculate \( \frac{du}{d\theta} \) and solve for \( d\theta \), replacing with \( du \) terms in the integral.
• Transform the entire integral in terms of \( u \) by substituting \( g(\theta) \) with \( u \).
• After integrating, substitute back the original variable to express the result in terms of the original function.

In the original exercise, \( u = \sin(5\theta) \) was chosen, simplifying the original integral \( \int \sin^6(5\theta) \cos(5\theta) \, d\theta \) into a more manageable form. This elegant technique allowed us to transform and eventually integrate the function, arriving at the final solution.