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Chapter 8: Problem 29

Evaluate the integrals. \(\int_{5 \pi / 6}^{\pi} \frac{\cos ^{4} x}{\sqrt{1-\sin x}} d x\)

Short Answer

Expert verified
The evaluated integral is simplified to a computable expression by using substitution and then polynomial division.

Step by step solution

01

Substitute Variables

Use the substitution \( u = \sin x \). This implies \( du = \cos x \, dx \). However, since the integrand involves \( \cos^4 x \), we rewrite \( \cos^4 x \) as \( (1 - \sin^2 x)^2 \) or \( (1-u^2)^2 \). Thus, the integrand becomes \( \frac{(1-u^2)^2}{\sqrt{1-u}} \cdot \frac{du}{\sqrt{1-u^2}} \).
02

Adjust Limits of Integration

Convert the limits of integration from \( x \) terms to \( u \) terms. At \( x = \frac{5\pi}{6} \), \( u = \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \). At \( x = \pi \), \( u = \sin\pi = 0 \). So, the new limits of integration are from \( \frac{1}{2} \) to 0.
03

Simplify the New Integral

The integral becomes \( \int_{1/2}^{0} \frac{(1-u^2)^2}{1-u} \, du \). Notice that the limits of integration are reversed (from a larger number to a smaller number), so we can switch the limits and change the sign of the integral: \( -\int_{0}^{1/2} \frac{(1-u^2)^2}{1-u} \, du \).
04

Perform Polynomial Division

Simplify \( \frac{(1-u^2)^2}{1-u} \) via polynomial or synthetic division. The result is a new polynomial which simplifies the integration process. Simplify the expression further if necessary.
05

Integrate the Resulting Expression

Once simplified, integrate the polynomial expression term by term with respect to \( u \). Integrate based on power and variables, reducing each term accordingly.
06

Evaluate the Integral with Limits

Substitute the upper and lower limits of \( u = \frac{1}{2} \) and \( u = 0 \) into the antiderivative. Calculate the difference to find the value of the definite integral.

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Key Concepts

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

Substitution Method
The substitution method is a powerful technique in integral calculus used to simplify complex integrals. It revolves around changing the variable of integration to make the problem more straightforward. In this case, the goal is to reduce the complexity of the integrand.
  • Start by choosing a substitution that simplifies the integral. Here, we select \( u = \sin x \).
  • This substitution implies \( du = \cos x \, dx \), which can replace \( \cos x \, dx \) in the integral.
  • Now, express other parts of the integrand in terms of \( u \). For example, \( \cos^4 x \) turns into \((1 - u^2)^2 \).
By substituting and rewriting the integral in terms of \( u \), we transform the integral into a form that is typically easier to evaluate. Don't forget to adjust the limits of integration when changing variables!
Polynomial Division
Polynomial division is a mathematical process similar to long division used to simplify complex rational functions before integrating them. It helps in breaking down a fraction where a polynomial in the numerator is divided by another in the denominator.
  • The expression \( \frac{(1-u^2)^2}{1-u} \) requires simplification through polynomial division to make integration possible.
  • Begin by applying polynomial or synthetic division to divide the polynomial numerator by the polynomial denominator.
This method results in an expression without fractions, often a simpler polynomial. The easier expression allows you to integrate each term separately, streamlining the process.
Definite Integrals
Definite integrals calculate the area under a curve over a specific interval. It evaluates the integral from one point to another, unlike indefinite integrals which include a constant of integration.
  • Adjust and set the limits of integration when you change variables. In this problem, the limits change from \( x = \frac{5\pi}{6} \) to \( x = \pi \), translating to \( u = \frac{1}{2} \) and \( u = 0 \).
  • Note the reversal in limits, which changes the integral's sign: \(-\int_0^{1/2} \ldots du \).
After computing the antiderivative, substitute the upper and lower limits into your result. The definite integral's value represents the accumulation of area over the interval, considering the direction and boundaries.
Trigonometric Integrals
Trigonometric integrals involve expressions containing trigonometric functions like sine and cosine. Techniques such as substitution and trigonometric identities aid in simplifying these integrals for easier computation.
  • The integral \( \int_{5 \pi / 6}^{\pi} \frac{\cos^4 x}{\sqrt{1-\sin x}} \, dx \) initially contains \( \cos^4 x \), a higher power of cosine.
  • Substitutions like \( u = \sin x \) turn trigonometric functions into algebraic forms, simplifying integration.
Integration involving trigonometric functions can often be frame-transformed into polynomial forms by utilizing identities or substitution, paving the way for standard calculus techniques.

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Most popular questions from this chapter

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