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Chapter 4: Problem 44

Evaluate the integral. $$ \int_{\pi / 4}^{\pi / 2} \sin 2 x d x $$

Short Answer

Expert verified
The integral can be evaluated as follows: first, apply the double angle identity, \(\sin2x = 2\sin x \cos x\), and rewrite the integral as \(\int_{\pi / 4}^{\pi / 2} 2 \sin x \cos x \, d x\). Then, find the antiderivative which is \((\sin x)^2 + C\), and finally, evaluate the antiderivative within the limits of integration, \(\pi / 4\) and \(\pi / 2\), to get the result: \(\int_{\pi / 4}^{\pi / 2} \sin 2 x \, d x = \frac{1}{2}\).

Step by step solution

01

Recall the identity for sin(2x)

Recall the double angle identity for sine, which states that: \[ \sin 2x = 2 \sin x \cdot \cos x \]
02

Apply the double angle identity to the integral

Replace sin(2x) with the identity we recalled and rewrite the integral as: \[ \int_{\pi / 4}^{\pi / 2} 2 \sin x \cos x \, d x \]
03

Compute the antiderivative

In order to evaluate the integral, we need to find the antiderivative of the function inside the integral: \[ \int 2 \sin x \cos x \, d x \] We are going to use the substitution method. Let's substitute \(u = \sin x\), then the differential of u is \(du = \cos x \, d x\). Now rewrite the integral in terms of u: \[ \int 2 u \, du \] Now integrate with respect to u: \[ u^2 + C \] Now, substitute back sin(x) for u: \[ (\sin x)^2 + C \]
04

Evaluate the definite integral

Now that we have the antiderivative, we need to evaluate it within the limits of integration, \(\pi / 4\) and \(\pi / 2\). Using the Fundamental Theorem of Calculus, we have: \[ \left[(\sin x)^2\right]_{\pi / 4}^{\pi / 2} \] Which means: \[ (\sin (\pi/2))^2 - (\sin (\pi/4))^2 \] Calculate the sine values: \[ (1)^2 - (\frac{\sqrt{2}}{2})^2 \]
05

Simplify the result

Simplify the expression: \[ 1 - \frac{1}{2} = \frac{1}{2} \] Thus, the result of the integral is: \[ \int_{\pi / 4}^{\pi / 2} \sin 2 x \, d x = \frac{1}{2} \]

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Key Concepts

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

Trigonometric Integrals
Understanding how to evaluate trigonometric integrals is essential for calculus students. These integrals involve trigonometric functions such as sine, cosine, and tangent. When faced with an integral like \(\int \sin 2x \, dx\), recognizing patterns and identities can simplify the problem drastically. For instance, applying the double angle identity can transform the original integrand into an equivalent expression that is easier to integrate. Moreover, trigonometric integrals often rely on substitution to further simplify the expression, after which you can find the antiderivative to evaluate the definite integral.

Double Angle Identity
The double angle identity for sine states that \(\sin 2x = 2 \sin x \cdot \cos x\). This identity allows us to break down more complex trigonometric functions into simpler ones that are easier to integrate. In the given problem, by applying the double angle identity, we rewrite the more complex \(\sin 2x\) as \(2 \sin x \cos x\), which is a straightforward product of sine and cosine functions. Recognizing and correctly using trigonometric identities is often a key step in solving trigonometric integrals.

Substitution Method
The substitution method, often referred to as u-substitution, is a technique for finding antiderivatives and evaluating definite integrals. It involves selecting a part of the integrand to be \(u\), which simplifies the integral into a form that is easier to manage. For example, in the exercise, setting \(u = \sin x\) and \(du = \cos x \, dx\) turns the integral \(\int 2 \sin x \cos x \, dx\) into \(\int 2 u \, du\), which is a basic integral that can be easily evaluated.

Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects differentiation and integration, providing a way to evaluate definite integrals. It states that if \(F(x)\) is an antiderivative of \(f(x)\), then \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\). In practice, once we find the antiderivative of the integrand, we can simply substitute the upper and lower bounds of the integral and subtract the results to find the area under the curve between those points.

Antiderivative
An antiderivative, also known as an indefinite integral, is a function \(F(x)\) such that \(F'(x) = f(x)\). When calculating definite integrals, the antiderivative allows us to determine the accumulation of quantity, such as area under a curve, between two points. In the context of the exercise, finding the antiderivative of \(2 \sin x \cos x\) is a crucial step before applying the Fundamental Theorem of Calculus to evaluate the definite integral.

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