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Chapter 6: Problem 40

Find or evaluate the integral. $$ \int \sin 3 \theta \sin 4 \theta d \theta $$

Short Answer

Expert verified
The evaluated integral is: $$ \int \sin 3 \theta \sin 4 \theta d\theta = \dfrac{1}{2}\left[\sin(\theta) - \dfrac{1}{7}\sin(7\theta)\right] + C $$

Step by step solution

01

Write down the given integral

The given integral is $$ \int \sin 3 \theta \sin 4 \theta d \theta $$
02

Apply the product-to-sum formula

Using the product-to-sum formula, we can rewrite the given integral as $$ \int \dfrac{1}{2}\left(\cos(3\theta - 4\theta) - \cos(3\theta + 4\theta)\right) d \theta $$ Simplify the integral: $$ \int \dfrac{1}{2}\left(\cos(-\theta) - \cos(7\theta)\right) d \theta $$
03

Evaluate the integral

Now we can split the integral and evaluate it: $$ \dfrac{1}{2}\int \left(\cos(-\theta) - \cos(7\theta)\right) d \theta = \dfrac{1}{2} \int \cos(-\theta) d \theta - \dfrac{1}{2} \int \cos(7 \theta) d \theta $$ Since \(\cos(-\theta) = \cos(\theta)\), we have $$ = \dfrac{1}{2} \int\cos(\theta) d \theta - \dfrac{1}{2} \int \cos(7 \theta) d \theta $$ Now find the antiderivatives: $$ = \dfrac{1}{2}\left[\sin(\theta) - \dfrac{1}{7}\sin(7\theta)\right] + C $$
04

Write the final answer

The evaluated integral is: $$ \int \sin 3 \theta \sin 4 \theta d\theta = \dfrac{1}{2}\left[\sin(\theta) - \dfrac{1}{7}\sin(7\theta)\right] + C $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product-to-Sum Identities
Understanding product-to-sum identities can greatly simplify the process of integrating trigonometric functions. These identities are particularly handy when you're dealing with products of sine and cosine. Instead of directly tackling the integral of a product
  • We transform the expression into a sum that is often easier to integrate.
  • For example, the integral of \( \sin(3\theta) \sin(4\theta) \) is not straightforward on its own.
  • However, by applying the product-to-sum formula, it becomes a combination of cosine terms:
    • \[ \sin(A)\sin(B) = \frac{1}{2} \left(\cos(A-B) - \cos(A+B)\right) \]
This identity recasts our original problem by introducing simpler trigonometric terms that are easier to integrate individually. By adopting this change, the apparent complexity is reduced to integrals of individual cosine functions.
Integration Techniques
Integration techniques include various methods to find the antiderivative and solve integrals. By learning these techniques, you improve your problem-solving toolbox. When faced with an integral function like \( \int \sin(3\theta) \sin(4\theta) d\theta \),
  • We separated the integral into two main parts by employing trigonometric identities.
  • This converted it into simpler terms: \[ \frac{1}{2} \left( \int \cos(-\theta) d\theta - \int \cos(7\theta) d\theta \right) \]
From here, each part can be individually integrated using basic antiderivative knowledge:
  • The integral of \( \cos(\theta) \) is straightforward, resulting in \( \sin(\theta) \).
  • Similarly, integrating \( \cos(k\theta) \) provides \( \frac{1}{k}\sin(k\theta) \).
These simple integrations add up to give comprehensive solutions to complex trigonometric integrals.
Definite and Indefinite Integrals
Differentiating between definite and indefinite integrals is crucial in calculus. Indefinite integrals are concerned with finding an antiderivative of a function, much like a reverse of differentiation. They don't have bounds and usually include a constant of integration, \( C \).
  • For \( \int \sin(3\theta) \sin(4\theta) d\theta \), the solved form is \[ \frac{1}{2}\left[\sin(\theta) - \frac{1}{7}\sin(7\theta)\right] + C \] where \( C \) indicates an indefinite integral.
  • This constant is essential because the antiderivative isn't unique, and \( C \) accounts for all possible vertical shifts of the solution.
Definite integrals, on the other hand, compute the area under a curve over a specified interval and don't include \( C \). The evaluation steps may use similar techniques, but the results are numerical values rather than functions. Understanding when and how to apply each type is key in calculus.

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Most popular questions from this chapter

Find or evaluate the integral. $$ \int \frac{x+3}{\sqrt{5-4 x-x^{2}}} d x $$

Find the area of the region under the graph of \(y=\frac{x^{3}}{x^{3}+1}\) on the interval \([0,2]\).

Use the Comparison Test to determine whether the integral is convergent or divergent by comparing it with the second integral. $$ \int_{1}^{\infty} \frac{2+\cos x}{\sqrt{x}} d x ; \int_{1}^{\infty} \frac{1}{\sqrt{x}} d x $$

Find or evaluate the integral. $$ \int \frac{d x}{x+1+\sqrt{x+1}} $$

Use the Comparison Test to determine whether the integral is convergent or divergent by comparing it with the second integral. $$ \int_{1}^{\infty} \frac{1}{1+x^{2}} d x ; \quad \int_{1}^{\infty} \frac{1}{x^{2}} d x $$
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