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Chapter 8: Problem 65

Use any method to evaluate the integrals. $$\int \frac{\tan ^{2} x}{\csc x} d x$$

Short Answer

Expert verified
\( \int \frac{\tan^2 x}{\csc x} dx = \sec x + \cos x + C \).

Step by step solution

01

Simplify the Integrand

Start by converting all trigonometric functions in the integrand into sine and cosine. Recall that \( \tan x = \frac{\sin x}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \). So the integrand \( \frac{\tan^2 x}{\csc x} \) becomes:\[\frac{\left(\frac{\sin^2 x}{\cos^2 x}\right)}{\frac{1}{\sin x}} = \frac{\sin^2 x}{\cos^2 x} \cdot \sin x = \frac{\sin^3 x}{\cos^2 x}\]
02

Substitute to Simplify Further

Notice that \( \frac{\sin^3 x}{\cos^2 x} \) can be made simpler by letting \( u = \cos x \), leading to \( du = -\sin x \, dx \). Rewrite \( \sin^3 x \) in terms of \( \cos x \) using this substitution:\[\sin^2 x = 1 - \cos^2 x = 1 - u^2\]So our integrand becomes:\[-\int \frac{(1 - u^2) \sin x}{u^2} du = -\int \frac{(1-u^2)}{u^2} d(-u)\]
03

Integrate with Respect to u

Now, split the fraction and integrate with respect to \( u \):\[-\int \left( \frac{1}{u^2} - 1 \right) du = -\left[ \int \frac{1}{u^2} du - \int 1 \, du \right]\]The integral \( \int \frac{1}{u^2} du \) is \( -\frac{1}{u} \), and \( \int 1 \, du \) is \( u \). Thus:\[-(-\frac{1}{u} - u) = \frac{1 + u^2}{u}\]
04

Substitute Back to x

Replace \( u \) back with \( \cos x \) to find the antiderivative in terms of \( x \):\[\frac{1 + \cos^2 x}{\cos x} = \frac{1}{\cos x} + \frac{\cos^2 x}{\cos x} = \sec x + \cos x\]
05

Apply Limits if Needed

If there are no specific limits given, then the indefinite integral becomes:\[\int \frac{\tan^2 x}{\csc x} dx = \sec x + \cos x + C\]where \( C \) is the constant of integration, indicating any constant can be added to this indefinite integral.

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Key Concepts

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

Trigonometric Identities
Trigonometric identities play a crucial role in simplifying complex integrals. They allow us to transform trigonometric functions into more manageable forms.
To begin solving integrals involving trigonometric functions, it's common to use basic identities:
  • The tangent function, \( \tan x \), can be expressed as \( \frac{\sin x}{\cos x} \).
  • The cosecant function, \( \csc x \), is the reciprocal of sine, \( \frac{1}{\sin x} \).
By substituting these identities into the integrand, we often make calculations simpler.
In our exercise, by transforming \( \tan^2 x \) and \( \csc x \) using their sine and cosine definitions, we can express the integrand entirely in terms of sine and cosine.
This step is vital because it transforms a seemingly complicated expression into a format that is easier to handle with integration techniques.
Integration Techniques
Integration techniques are strategies used to solve integrals that are not immediately straightforward.
One powerful method is substitution, which simplifies integrals by transforming variables.
When faced with an integral like \( \frac{\sin^3 x}{\cos^2 x} \), substituting a suitable variable like \( u = \cos x \) can greatly simplify the process.
This involves calculating the differential \( du = -\sin x \, dx \), which helps relate \( dx \) and \( du \) in the integral.
  • Upon substitution, the integrand changes, and integration can be performed with respect to \( u \) rather than \( x \).
  • This step often leads to simpler algebraic expressions, like fractions that can be split into separate integrals.
  • After integrating in terms of \( u \), substituting back gives the antiderivative in terms of the original variable.
Mastering substitution requires practice but is invaluable for handling a wide range of indefinite integrals.
Indefinite Integrals
Indefinite integrals are fundamental in calculus, representing the family of functions whose derivative gives the function under the integral.
The notation \( \int f(x) \, dx \) seeks a function, \( F(x) \), such that \( F'(x) = f(x) \).
  • The indefinite integral does not specify limits, meaning it represents a broad category of functions inclusive of all possible constants.
  • Constants are accounted for with \( C \), the constant of integration, reflecting any constant number that may have been part of the original function鈥檚 derivative.
In the exercise, the result \( \sec x + \cos x + C \) indicates that there could be multiple antiderivatives, each differing by a constant \( C \).
This is critical because it emphasizes the general solution without specific limits, applicable to any value of \( x \).
Being comfortable with indefinite integrals is essential, as they form the foundation for understanding more complex topics in calculus.

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Most popular questions from this chapter

Evaluate the integrals in Exercises \(39-50\). $$\int \frac{1}{x^{6}\left(x^{5}+4\right)} d x$$

Use a CAS to explore the integrals for various values of \(p\) (include noninteger values). For what values of \(p\) does the integral converge? What is the value of the integral when it does converge? Plot the integrand for various values of \(p.\) $$\int_{e}^{\infty} x^{p} \ln x d x$$

a. Use a CAS to evaluate $$\int_{0}^{\pi / 2} \frac{\sin ^{n} x}{\sin ^{n} x+\cos ^{n} x} d x$$ where \(n\) is an arbitrary positive integer. Does your CAS find the result? b. In succession, find the integral when \(n=1,2,3,5,\) and 7 Comment on the complexity of the results. c. Now substitute \(x=(\pi / 2)-u\) and add the new and old integrals. What is the value of $$\int_{0}^{\pi / 2} \frac{\sin ^{n} x}{\sin ^{n} x+\cos ^{n} x} d x ?$$ This exercise illustrates how a little mathematical ingenuity solves a problem not immediately amenable to solution by a CAS.

Use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. $$\int_{1}^{\infty} \frac{e^{x}}{x} d x$$

The function $$\operatorname{erf}(x)=\int_{0}^{x} \frac{2 e^{-t^{2}}}{\sqrt{\pi}} d t$$ called the error function, has important applications in probability and statistics. a. Plot the error function for \(0 \leq x \leq 25\) b. Explore the convergence of $$\int_{0}^{\infty} \frac{2 e^{-t^{2}}}{\sqrt{\pi}} d t$$ If it converges, what appears to be its value? You will see how to confirm your estimate in Section \(14.4,\) Exercise \(41 .\)
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