/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 27

Evaluate the following integrals. $$\int \sin ^{4} \frac{x}{2} d x$$

Short Answer

Expert verified
Question: Find the integral of the function $$\int \sin ^{4} \frac{x}{2} d x$$. Answer: The integral of the function $$\int \sin ^{4} \frac{x}{2} d x$$ is given by $$\frac{3}{8} x - \frac{1}{2} \sin x + \frac{1}{16} \sin (2x) + C$$ where C is the constant of integration.

Step by step solution

01

Triple-Angle Formula of Sine

We will use the triple-angle formula of sine to rewrite the integrand: $$\sin^4 (\frac{x}{2}) = \left(\frac{1 - \cos (2 \cdot (\frac{x}{2}))}{2}\right)^2$$ Now our integral becomes, $$\int \left(\frac{1 - \cos x}{2}\right)^2 d x$$
02

Expand the integrand

We will expand the integrand and rewrite the integral: $$\int \left(\frac{1 - \cos x}{2}\right)^2 d x = \int \left(\frac{1}{4} - \frac{1}{2} \cos x + \frac{1}{4} \cos^2 x\right) d x$$
03

Use the half-angle identity

We will use a half-angle identity to simplify the integral: $$\cos^2 x = \frac{1 + \cos (2x)}{2}$$ So our integral becomes, $$\int \left(\frac{1}{4} - \frac{1}{2} \cos x + \frac{1}{8} + \frac{1}{8} \cos (2x)\right) d x = \int \left(\frac{3}{8} - \frac{1}{2} \cos x + \frac{1}{8} \cos (2x)\right) d x$$
04

Integrate each term

We will now integrate each term in the integrand: $$\int \left(\frac{3}{8} - \frac{1}{2} \cos x + \frac{1}{8} \cos (2x)\right) d x = \frac{3}{8} \int 1 d x - \frac{1}{2} \int \cos x d x + \frac{1}{8} \int \cos (2x) d x$$
05

Evaluate the integrals

Now, we can evaluate each integral: $$\frac{3}{8} \int 1 d x - \frac{1}{2} \int \cos x d x + \frac{1}{8} \int \cos (2x) d x = \frac{3}{8} x - \frac{1}{2} \sin x + \frac{1}{16} \sin (2x) + C$$ where C is the constant of integration. So the final result is: $$\int \sin ^{4} \frac{x}{2} d x = \frac{3}{8} x - \frac{1}{2} \sin x + \frac{1}{16} \sin (2x) + C$$

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Key Concepts

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

Triple-Angle Formula
The triple-angle formula is a useful tool when dealing with trigonometric functions involving angles that are three times another angle. Specifically, for the sine function, the triple-angle formula is expressed as follows:
\[\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta) \]
This formula helps in simplifying expressions where you encounter powers or multiples of sine. For instance, if you are faced with a sine expression such as \(\sin^4(\frac{x}{2})\), you can handle it by decomposing the power using known trigonometric identities and formulas, including the triple-angle formula as seen in a step-by-step integration problem.
Half-Angle Identity
The half-angle identity is vital for simplifying expressions involving trigonometric functions raised to a power. For cosine, the half-angle identity is given by:
\[\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}\]
This formula is particularly useful in integration, as it allows conversion of a squared trigonometric function into a simpler form integrable in a straightforward manner. By using this identity, one can transform integrands like \(\cos^2(x)\) into a more manageable polynomial-like expression, thus simplifying the computational process in integral calculus.
Sine Function
The sine function, denoted as \(\sin(x)\), is a fundamental trigonometric function describing the vertical position on the unit circle of an angle \(x\). Its properties include:
  • Periodicity with a period of \(2\pi\).
  • Range between -1 and 1.
  • Symmetrical properties, such as \(\sin(-x) = -\sin(x)\).

These features are critical when evaluating integrals, especially those involving power functions of sine. Knowing how the sine function behaves simplifies not only the algebraic manipulations but also the integration steps involving sine terms. This understanding can convert complex integrals into concise and easily solvable integrations.
Cosine Function
Like the sine function, the cosine function, represented as \(\cos(x)\), measures the horizontal position of an angle \(x\) on the unit circle. It shares several characteristics with sine:
  • Also periodic with a period of \(2\pi\).
  • Range lies between -1 and 1.
  • Even function property: \(\cos(-x) = \cos(x)\).

These properties make cosine particularly valuable when paired with its identity transformations during integration. The cosine function allows for expressing more complex trigonometric identities and plays a vital role in the computation of integrals. By leveraging its periodic and symmetry properties, integrals involving the cosine function can be evaluated with greater simplicity and accuracy.

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Most popular questions from this chapter

Another form of \(\int \sec x \, d x\) a. Verify the identity sec \(x=\frac{\cos x}{1-\sin ^{2} x}\) b. Use the identity in part (a) to verify that \(\int \sec x \, d x=\frac{1}{2} \ln \left|\frac{1+\sin x}{1-\sin x}\right|+C\)

Determine whether the following integrals converge or diverge. $$ \int_{1}^{\infty} \frac{\sin ^{2} x}{x^{2}} d x $$

Evaluate the following integrals. Assume a and b are real numbers and \(n\) is a positive integer. \(\int x^{n} \sin ^{-1} x d x\) (Hint. integration by parts.)$

Trapezoid Rule and concavity Suppose \(f\) is positive and its first two derivatives are continuous on \([a, b] .\) If \(f^{\prime \prime}\) is positive on \([a, b]\) then is a Trapezoid Rule estimate of \(\int_{a}^{b} f(x) d x\) an underestimate or overestimate of the integral? Justify your answer using Theorem 8.1 and an illustration.

Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals. $$\int \frac{\left(e^{3 x}+e^{2 x}+e^{x}\right)}{\left(e^{2 x}+1\right)^{2}} d x$$
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