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Chapter 5: Problem 18

\(5-18\) Find the general indefinite integral. $$\int \frac{\sin 2 x}{\sin x} d x$$

Short Answer

Expert verified
The general indefinite integral is \( 2 \sin x + C \).

Step by step solution

01

Simplify the Integrand

The given integral is \( \int \frac{\sin 2x}{\sin x} \, dx \). Using the double angle identity, \( \sin 2x = 2 \sin x \cos x \), we can rewrite the integrand:\[ \frac{\sin 2x}{\sin x} = \frac{2 \sin x \cos x}{\sin x} = 2 \cos x \]
02

Rewrite the Integral

Substitute the simplified expression into the integral:\[ \int \frac{\sin 2x}{\sin x} \, dx = \int 2 \cos x \, dx \]
03

Integrate the Expression

The integral of \( \cos x \) is \( \sin x \). So, integrating \( 2 \cos x \) gives:\[ \int 2 \cos x \, dx = 2 \sin x + C \]where \( C \) is the constant of integration.

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Key Concepts

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

Trigonometric Identities
To solve integrals involving trigonometric functions, we often rely on trigonometric identities. These identities are fundamental relationships with specific angles or functions.
In the context of this exercise, the double angle identity for sine is used. The double angle identity for sine is: \( \sin 2x = 2 \sin x \cos x \).
This identity helps break down complex functions into simpler ones. By expressing \( \sin 2x \) in terms of \( \sin x \) and \( \cos x \), it becomes easier to manipulate and integrate the function. Understanding and recognizing these identities is crucial for simplifying trigonometric problems.
Integrand Simplification
The process of solving indefinite integrals often involves simplifying the integrand.
In this example, the integrand \( \frac{\sin 2x}{\sin x} \) can initially seem complex. However, using trigonometric identities aids in simplifying it. You notice that \( \sin x \) terms in both the numerator and denominator allow for cancellation.
Thus, the expression is reduced to \( 2 \cos x \). Simplification is essential because it reveals a pathway to solve integrals that might otherwise appear daunting. Always check for common factors or identities that could simplify the expression.
Double Angle Identity
The double angle identity plays a pivotal role in transforming this integral into a more manageable form. It's necessary to recall that \( \sin 2x = 2 \sin x \cos x \). This identity is particularly useful because it directly connects angles and functions that aren't immediately obvious.
By implementing the double angle formula, the integration becomes straightforward because the once complex looking integrand \( \frac{\sin 2x}{\sin x} \) simplifies to \( 2 \cos x \). Renewal of perspective through identities often makes difficult problems easier to solve.
Constant of Integration
When dealing with indefinite integrals, one must always remember to include the constant of integration. Indefinite integrals generally produce a family of functions.
Adding a constant \( C \) to the antiderivative \( 2 \sin x \) acknowledges all possible vertical shifts of the function.
Integration reverses differentiation, so the constant of integration becomes a crucial component that represents any vertical displacement that was derivatively neutral in the original function. Always include it to mirror the general solution accurately.

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Most popular questions from this chapter

The error function $$\operatorname{erf}(\mathrm{x})=\frac{2}{\sqrt{\pi}} \int_{0}^{\mathrm{x}} \mathrm{e}^{-\mathrm{t}^{2}} \mathrm{dt}$$ is used in probability, statistics, and engineering. $$\begin{array}{l}{\text { (a) Show that } \int_{a}^{b} e^{-t^{2}} d t=\frac{1}{2} \sqrt{\pi}[\operatorname{erf}(b)-\operatorname{erf}(a)] .} \\\ {\text { (b) Show that the function } y=e^{x^{2}} \operatorname{erf}(x) \text { satisfies the differ- }} \\ {\text { ential equation } y^{\prime}=2 x y+2 / \sqrt{\pi}}\end{array}$$

The current in a wire is defined as the derivative of the charge: \(I(t)=Q^{\prime}(t) .\) (See Example 3 in Section \(3.7 . )\) What does \(\int_{a}^{b} I(t)\) dt represent?

The marginal cost of manufacturing \(x\) yards of a certain fabric is \(C^{\prime}(x)=3-0.01 x+0.00006 x^{2}\) (in dollars per yard). Find the increase in cost if the production level is raised from 2000 yards to 4000 yards.

\(21-44\) Evaluate the integral. $$\int_{1}^{2} \frac{y+5 y^{7}}{y^{3}} d y$$

If \(x\) is measured in meters and \(f(x)\) is measured in newtons, what are the units for \(\int_{0}^{100} f(x) d x ?\)
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