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Magazine Chapter 28: Problem 17
Integrate each of the given functions. $$\int_{0}^{\pi / 4} \frac{\tan x}{\cos ^{4} x} d x$$
Short Answer
Expert verified
The integral evaluates to \( \frac{3}{4} \).
Step by step solution
01
Understanding the Given Integral
The integral to evaluate is \( \int_{0}^{\pi / 4} \frac{\tan x}{\cos^4 x} \ dx \). The function involves the tangent and cosine functions raised to a power. The goal is to find an antiderivative that simplifies this expression.
02
Simplifying the Integrand
Rewrite the integrand \( \frac{\tan x}{\cos^4 x} \) in terms of sine and cosine. Recall that \( \tan x = \frac{\sin x}{\cos x} \). Hence, the integrand becomes:\[\frac{\sin x}{\cos x} \cdot \frac{1}{\cos^4 x} = \frac{\sin x}{\cos^5 x}.\]
03
Introducing a Suitable Substitution
Use the substitution \( u = \cos x \), which implies \( du = -\sin x \, dx \) or \( \sin x \, dx = -du \). Replace \( \sin x \, dx \) in the integral with \(-du\):\[\int \frac{\sin x}{\cos^5 x} \ dx = \int \frac{-du}{u^5}.\]
04
Integrating with Respect to \( u \)
The integral \( \int \frac{-1}{u^5} \, du \) can be evaluated using the power rule for integration, \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) (for \( n eq -1 \)):\[\int \frac{-1}{u^5} \, du = \int -u^{-5} \, du = \left[-\frac{u^{-4}}{-4}\right] + C = \frac{1}{4u^4} + C.\]
05
Back-Substitute \( u \) to \( x \)
Substitute back \( u = \cos x \) into the antiderivative:\[\frac{1}{4u^4} = \frac{1}{4(\cos x)^4}.\]
06
Evaluating the Definite Integral
Evaluate \( \left. \frac{1}{4(\cos x)^4} \right|_{0}^{\pi/4} \):- At \( x = 0 \), \( \cos 0 = 1 \), so \( \frac{1}{4(1)^4} = \frac{1}{4} \).- At \( x = \pi/4 \), \( \cos(\pi/4) = \frac{\sqrt{2}}{2} \), so \( \frac{1}{4((\frac{\sqrt{2}}{2})^4)} = \frac{1}{4(\frac{1}{4})} = 1 \).Thus, the definite integral is:\[1 - \frac{1}{4} = \frac{3}{4}.\]
07
Conclusion
The value of the definite integral \( \int_{0}^{\pi / 4} \frac{\tan x}{\cos^4 x} \ dx \) is \( \frac{3}{4} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integral
A definite integral gives the accumulation of a quantity over a certain interval. When we calculate a definite integral, we are computing the net area under a curve from one point to another along the x-axis.
In our problem, we are tasked with finding the definite integral from 0 to \(\frac{\pi}{4}\) for the function \( \frac{\tan x}{\cos ^4 x} \).To solve a definite integral, we follow a few key steps:
In our problem, we are tasked with finding the definite integral from 0 to \(\frac{\pi}{4}\) for the function \( \frac{\tan x}{\cos ^4 x} \).To solve a definite integral, we follow a few key steps:
- Evaluate the indefinite integral, also known as the antiderivative, of the given function.
- Substitute the upper and lower limits of the integral into the antiderivative.
- Find the difference between these two values to reach the final result.
Trigonometric Substitution
Trigonometric substitution is a useful technique for simplifying integrals involving trigonometric functions. In our exercise, we have \( \tan x \) and \( \cos^4 x \), which can be challenging to integrate directly.
The trick is to use trigonometric identities to transform these functions into simpler expressions. In this context, the substitution \( u = \cos x \) is employed. This results in \( du = -\sin x \, dx \), thus making the integral easier to handle:
The trick is to use trigonometric identities to transform these functions into simpler expressions. In this context, the substitution \( u = \cos x \) is employed. This results in \( du = -\sin x \, dx \), thus making the integral easier to handle:
- Express \( \tan x \) as \( \frac{\sin x}{\cos x} \).
- Convert the integrand \( \frac{\sin x}{\cos^5 x} \) appropriately as \( -\frac{du}{u^5} \) by recognizing the connection between \( \sin x \, dx \) and \( du \).
Power Rule for Integration
The power rule for integration is a fundamental principle in integral calculus. It offers a straightforward way to find antiderivatives of power functions.
The rule states that if \( f(x) = x^n \) where \( n eq -1 \), the antiderivative is given by \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).In the exercise, after employing trigonometric substitution, we transformed the integral to \( \int -u^{-5} \, du \), which fits neatly into the power rule format:
The rule states that if \( f(x) = x^n \) where \( n eq -1 \), the antiderivative is given by \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).In the exercise, after employing trigonometric substitution, we transformed the integral to \( \int -u^{-5} \, du \), which fits neatly into the power rule format:
- Apply the power rule to find \( \int -u^{-5} \, du = \frac{u^{-4}}{4} + C = \frac{1}{4u^4} + C \).
- Remember to back-substitute the original variable to express the antiderivative in terms of \( x \).
