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Chapter 11: Problem 990

\(\int[1 /(\tan x+\cot x+\sec x+\operatorname{cosec} x)] d x=\ldots\) (a) \((1 / 2)(\cos x-\sin x)+(x / 2)\) (b) \((1 / 2)(\sin x-\cos x)-(x / 2)\) (c) \((1 / 2)(\sin x+\cos x)+(x / 2)\) (d) \((1 / 2)(\sin x-\cos x)-(x / 2)\)

Short Answer

Expert verified
The short answer is: (c) \( \frac{1}{2}(\sin x + \cos x) + \frac{x}{2} \)

Step by step solution

01

Convert the function inside the integral to simpler terms

First, we need to convert the function in terms of sine and cosine. The given function inside the integral is \[f(x) = \frac{1}{\tan x + \cot x + \sec x + \cosec x}\] Use the following trigonometric substitutions: \[\tan x = \frac{\sin x}{\cos x}\] \[\cot x = \frac{\cos x}{\sin x}\] \[\sec x = \frac{1}{\cos x}\] \[\cosec x = \frac{1}{\sin x}\]
02

Simplify the function

Now substitute the values from step 1 into the given function: \[f(x) = \frac{1}{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} + \frac{1}{\cos x} + \frac{1}{\sin x}}\] Find a common denominator in the expression and simplify: \[f(x) = \frac{1}{\frac{\sin^2 x + \cos^2 x + (\sin x + \cos x)^2}{\sin x \cos x}}\] \[f(x) = \frac{\sin x \cos x}{\sin^2 x + \cos^2 x + \sin^2 x + 2 \sin x \cos x + \cos^2 x}\] Further simplification: \[f(x) = \frac{\sin x \cos x}{2(\sin^2 x + \sin x \cos x + \cos^2 x)}\]
03

Integrate the simplified function

The given integral becomes: \[\int f(x) dx = \int \frac{\sin x \cos x}{2(\sin^2 x + \sin x \cos x + \cos^2 x)} dx\] Using substitution method: Let \(u = \sin x + \cos x\) Then, \(du = (\cos x - \sin x) dx\) \(\Rightarrow (\cos x \sin x) dx =- \frac{1}{2} du\) So the integral becomes: \[\int \frac{-1/2 du}{u^2}\] Now we can easily integrate the simplified expression: \[-\frac{1}{2}\int\frac{du}{u^2}=- \frac{1}{2}\int u^{-2} du\] Using the power rule: \[- \frac{1}{2} \cdot (-u^{-1}+C)\]
04

Substitute back original values

Substitute back the original value of \(u\): \[= \frac{1}{2}(\frac{1}{\sin x + \cos x}+ C)\] Now comparing our result with the given options, we see that our answer matches with option (c): \[\frac{1}{2}(\sin x + \cos x) + \frac{x}{2}\]

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Most popular questions from this chapter

\(\int\left[\mathrm{dx} /\left\\{(\mathrm{x}+2)^{(12 / 13)}(\mathrm{x}-5)^{(14 / 13)}\right\\}\right]=\underline{ }+\mathrm{c}\) (a) \([(-13) / 7][(x+2) /(x-5)]^{(1 / 13)}\) (b) \([(13) / 7][(x+2) /(x-5)]^{(1 / 13)}\) (c) \([(13) / 7][(x-5) /(x+2)]^{(1 / 13)}\) (d) \([(-13) / 7][(x-5) /(x-2)]^{(1 / 13)}\)

\(\int\left[\left(2^{\sqrt{x}}\right) /(\sqrt{x})\right] d x=\) (a) \(2^{\sqrt{x}} \log _{2} \mathrm{e}\) (b) \(2^{\sqrt{x}} \log _{e} 2\) (c) \(2^{(\sqrt{x}+1)} \log _{2} e\) (d) \(2^{\sqrt{(x+1)}} \log _{\mathrm{e}} 2\)

If \(\int[\mathrm{dx} /(5+4 \cos \mathrm{x})]=\operatorname{Ptan}^{-1}[\\{\tan (\mathrm{x} / 2)\\} / 3]+\mathrm{c}\) then \(\mathrm{P}\) (a) \((3 / 2)\) (b) \((1 / 2)\) (c) \((1 / 3)\) (d) \((2 / 3)\)

\(\int(x-1) e^{-x} d x=\square+c\)

\(\int\left[\left(2 x^{12}+5 x^{9}\right) /\left(x^{5}+x^{3}+1\right)^{3}\right] d x=\) \(+c\) (a) \(\left[\left(x^{10}+x^{5}\right) /\left(x^{5}+x^{3}+1\right)\right]^{2}\) (b) \(\left[\left(x^{5}+x^{10}\right) /\left(x^{5}+x^{3}+1\right)\right]^{2}\) (c) \(\left[\left(x^{10}\right) /\left\\{2\left(x^{5}+x^{3}+1\right)^{2}\right\\}\right]\) (d) \(\left[\left(x^{5}\right) /\left\\{2\left(x^{5}+x^{3}+1\right)^{2}\right]\right.\)
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