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Chapter 7: Problem 1

State the half-angle identities used to integrate \(\sin ^{2} x\) and \(\cos ^{2} x\)

Short Answer

Expert verified
Answer: The integrated expressions are: \(\int \sin^2x \,dx = \frac{1}{2}(x-\frac{\sin(2x)}{2}) + C_1\), \(\int \cos^2x \,dx = \frac{1}{2}(x+\frac{\sin(2x)}{2}) + C_2\).

Step by step solution

01

Identify the Half-Angle Identities

The half-angle identities for \(\sin^2x\) and \(\cos^2x\) are given by the following formulas: $$\sin^2x = \frac{1 - \cos(2x)}{2}$$ $$\cos^2x = \frac{1 + \cos(2x)}{2}$$
02

Substitute the Half-Angle Identities into the Integrals

Replace \(\sin^2x\) and \(\cos^2x\) with their respective half-angle identities in the integrals: $$\int \sin^2x \,dx = \int \frac{1 - \cos(2x)}{2} \,dx$$ $$\int \cos^2x \,dx = \int \frac{1 + \cos(2x)}{2} \,dx$$
03

Integrate the Expressions

Now, integrate each expression: For \(\int \sin^2x \,dx\): $$\int \frac{1 - \cos(2x)}{2} \,dx = \frac{1}{2} \int (1 - \cos(2x)) \,dx$$ Now, integrate each term (with respect to x): $$\int 1 \,dx = x$$ $$\int -\cos(2x) \,dx = -\frac{1}{2} \sin(2x)$$ Combine the integrated terms: $$\frac{1}{2}(x - \frac{1}{2} \sin(2x)) + C_1 = \frac{1}{2}(x-\frac{\sin(2x)}{2}) + C_1$$ For \(\int \cos^2x \,dx\): $$\int \frac{1 + \cos(2x)}{2} \,dx = \frac{1}{2} \int (1 + \cos(2x)) \,dx$$ Now, integrate each term (with respect to x): $$\int 1 \,dx = x$$ $$\int \cos(2x) \,dx = \frac{1}{2} \sin(2x)$$ Combine the integrated terms: $$\frac{1}{2}(x + \frac{1}{2} \sin(2x)) + C_2 = \frac{1}{2}(x+\frac{\sin(2x)}{2}) + C_2$$ So, we have: $$\int \sin^2x \,dx = \frac{1}{2}(x-\frac{\sin(2x)}{2}) + C_1$$ $$\int \cos^2x \,dx = \frac{1}{2}(x+\frac{\sin(2x)}{2}) + C_2$$

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Key Concepts

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

Trigonometric Integration
Trigonometric integration involves integrating functions that contain trigonometric expressions, like sine or cosine. This type of integration can initially seem complicated due to the oscillating nature of trigonometric functions. However, a powerful tool called the half-angle identities simplifies these tasks significantly.
  • Half-angle identities are essential because they transform trigonometric expressions into simpler forms.
  • For instance, the half-angle identities for \(ackslash\sin^2x\) and \(ackslash\cos^2x\) are \(\sin^2x = \frac{1-ackslash\cos(2x)}{2}\) and \(ackslash\cos^2x = \frac{1+ackslash\cos(2x)}{2}\).
By substituting these identities into the integral, you convert a trigonometric function into an algebraic expression that is easier to integrate.This method is widely used because it simplifies calculations and reduces errors in manual integration.
Calculus
Calculus is the mathematical study of continuous change, fundamentally based around differentiation and integration. Integration, specifically, deals with finding the area under a curve or the accumulation of quantities. In simple terms, integration helps us find the whole or total when given a part or rate of change.
  • In the context of the original exercise, calculus is applied to find the integrals of \(\sin^2x\) and \(\cos^2x\).
  • These integrals are calculated using the half-angle identities to transform the trigonometric functions into simpler expressions.
This transformation enables us to apply basic calculus principles, like the antiderivative, more effectively.These manipulations are vital as they bridge the gap between trigonometric functions and polynomial expressions that are easier to work with.
Definite Integrals
Definite integrals are a concept in calculus where we evaluate the integral of a function between two specific limits. While the original exercise concerns indefinite integrals, understanding definite integrals provides a complete picture of integration.
  • A definite integral calculates the net area between the function and the x-axis over a specific interval.
  • It gives a numeric value that represents this area, as opposed to the indefinite integral, which represents a general family of functions.
To apply the definite integral to our earlier trigonometric example, once you have the indefinite integral, you use the upper and lower limits to evaluate it.The definite integral of \(\sin^2x\) or \(\cos^2x\) within a certain interval would give an actual area result, useful in various applications within physics and engineering.

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Most popular questions from this chapter

Use numerical methods or a calculator to approximate the following integrals as closely as possible. $$\int_{0}^{\infty} \frac{\sin ^{2} x}{x^{2}} d x=\frac{\pi}{2}$$

Decaying oscillations Let \(a>0\) and \(b\) be real numbers. Use integration to confirm the following identities. a. \(\int_{0}^{\infty} e^{-a x} \cos b x d x=\frac{a}{a^{2}+b^{2}}\) b. \(\int_{0}^{\infty} e^{-a x} \sin b x d x=\frac{b}{a^{2}+b^{2}}\)

An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution \(u=\tan (x / 2)\) or \(x=2 \tan ^{-1} u .\) The following relations are used in making this change of variables. $$A: d x=\frac{2}{1+u^{2}} d u \quad B: \sin x=\frac{2 u}{1+u^{2}} \quad C: \cos x=\frac{1-u^{2}}{1+u^{2}}$$ $$\text { Evaluate } \int \frac{d x}{1-\cos x}$$

Refer to the summary box (Partial Fraction Decompositions) and evaluate the following integrals. $$\int \frac{2}{x\left(x^{2}+1\right)^{2}} d x$$

Find the volume of the described solid of revolution or state that it does not exist. The region bounded by \(f(x)=(x+1)^{-3 / 2}\) and the \(y\) -axis on the interval (-1,1] is revolved about the line \(x=-1.\)
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