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Chapter 5: Problem 12

Evaluate the integrals. $$\int_{0}^{\pi / 3} 4 \frac{\sin u}{\cos ^{2} u} d u.$$

Short Answer

Expert verified
The definite integral evaluates to \( 4 \).

Step by step solution

01

Identify the Integral Type

The given integral is \( \int_{0}^{\pi/3} 4 \frac{\sin u}{\cos^2 u} \, du \). This is a definite integral involving a trigonometric function. Notice the integrand \( \frac{\sin u}{\cos^2 u} \) can be expressed in terms of a fundamental trigonometric identity.
02

Simplify the Integrand

Recognize that \( \frac{\sin u}{\cos^2 u} \) can be rewritten as \( \tan u \sec u \). This is because \( \tan u = \frac{\sin u}{\cos u} \) and \( \sec u = \frac{1}{\cos u} \). Therefore, the integral becomes \( \int 4 \tan u \sec u \, du \).
03

Find the Antiderivative

The integral of \( \tan u \sec u \) is a standard form. Recall that the antiderivative of \( \sec u \tan u \) is \( \sec u \). So, \( \int 4 \tan u \sec u \, du = 4 \sec u + C \), where \( C \) is the constant of integration.
04

Evaluate the Definite Integral

Now apply the limits from \( 0 \) to \( \frac{\pi}{3} \). This means evaluating \( 4 \sec u \) at these limits. Calculate \( 4 \sec \left( \frac{\pi}{3} \right) - 4 \sec(0) \).

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Key Concepts

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

Trigonometric Integrals
Trigonometric integrals like \(\int_{0}^{\pi/3} 4 \frac{\sin u}{\cos^2 u} \, du\) involve integrating functions that contain trigonometric identities or ratios.
  • These are common in calculus because they appear in various scientific and engineering contexts.
  • Recognizing trigonometric identities is crucial to simplifying these integrals. For instance, knowing that \( \tan u = \frac{\sin u}{\cos u} \) and \( \sec u = \frac{1}{\cos u} \) allows us to rewrite more complex expressions in simpler forms.
In this exercise, simplifying the integral involves converting \( \frac{\sin u}{\cos^2 u} \) into \( \tan u \sec u \). This transformation helps find the antiderivative more straightforwardly, aiding in the integration process, especially when involving definite integrals over specific intervals.
Antiderivatives
Understanding antiderivatives is a fundamental part of solving integrals. The antiderivative of a function is essentially its "opposite" operation to differentiation.
  • In simple terms, if you know a function's derivative, the antiderivative helps you "reverse" the process.
  • In this context, identifying that the antiderivative of \( \sec u \tan u \) is known enables straightforward calculation.
Once simplified to \( \int 4 \tan u \sec u \, du \), knowing that \( \int \tan u \sec u \, du = \sec u \) allows you to express the antiderivative as \( 4 \sec u + C \). This step is critical because it sets up the function for evaluation within specific bounds, completing the process of finding the definite integral.
Evaluating Limits
When dealing with definite integrals such as \(\int_0^{\pi/3} 4 \frac{\sin u}{\cos^2 u} \, du\), evaluating limits involves computing the difference between the antiderivative at the upper and lower bounds.
  • First, evaluate the antiderivative function at \( u = \frac{\pi}{3} \) where \( \sec(\frac{\pi}{3}) = 2 \).
  • Then at \( u = 0 \) where \( \sec(0) = 1 \).
By calculating \( 4 \sec(\frac{\pi}{3}) - 4 \sec(0) \), you find the value by substituting back into the antiderivative: \[ 4 \cdot 2 - 4 \cdot 1 = 8 - 4 = 4 \]This result, 4, gives the area under the curve \( \frac{\sin u}{\cos^2 u} \) from 0 to \( \frac{\pi}{3} \), thus completing the evaluation.

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