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Chapter 11: Problem 889

\(\int[\mathrm{dx} /(1+\tan \mathrm{x})]=\underline{ }+\mathrm{c}\) (a) \(\log |\sec x+\tan x|\) (b) \(2 \sec ^{2}(\mathrm{x} / 2)\) (c) \(\log \mid \mathrm{x}+\sin \mathrm{x}\) (d) \((1 / 2)[x+\log |\sin x+\cos x|]\)

Short Answer

Expert verified
None of the given options is correct.

Step by step solution

01

Set up the integral

Start by writing out the integral. In this case, the function to be integrated is \( \frac{dx}{1+\tan x} \).
02

Try a substitution

Set \( u = \tan x \), which implies \( du = \sec^2 x \, dx \). Substitute these values into the integral, giving \( \int \frac{1}{1 + u} \, du \).
03

Recognize a standard integral

This integral is a standard one – it is the integral of \( \frac{1}{1 + u} \), which is \( \ln |1+u| \).
04

Replace \(u\) with original value

Now that the integral has been solved, replace \( u \) with the original expression in terms of \( x \), i.e., \( \tan x \). So the antiderivative of \( \frac{1}{1 + \tan x} \) is \( \ln|1 + \tan x| \).
05

Return to Options

Compare this with the options provided. (a) \(\log |\sec x+\tan x|\) (b) \(2 \sec ^{2}(\mathrm{x} / 2)\) (c) \(\log \mid \mathrm{x}+\sin \mathrm{x}\) (d) \((1 / 2)[x+\log |\sin x+\cos x|]\) From these options, none of them matches the solution, \( \ln|1 + \tan x| \). This means that none of the options (a,b,c,d) is the correct answer to the given integral.

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Most popular questions from this chapter

If \(\int\left(x^{30}+x^{20}+x^{10}\right)\left(2 x^{20}+3 x^{10}+6\right)^{(1 / 10)} d x\) \(=\mathrm{k}\left(2 \mathrm{x}^{30}+3 \mathrm{x}^{20}+6 \mathrm{x}^{10}\right)^{(11 / 10)}+\mathrm{c}\) then \(\mathrm{k}=\) (c) \((1 / 66)\) (a) \((1 / 60)\) (b) \(-(1 / 60)\) (c) \(-(1 / 66)\)

If \(\int\left[\left(3^{x}-1\right) /\left(3^{x}+1\right)\right] d x=k \log \left|3^{(x / 2)}+3^{-(x / 2)}\right|+c\) then \(k=\) (a) \(\log _{3} \mathrm{e}\) (b) \(\log _{\mathrm{e}} 3\) (c) \(2 \log _{3} \mathrm{e}\) (d) \(2 \log _{\mathrm{e}} 3\)

If \(\int\left[\left(2 \mathrm{e}^{\mathrm{x}}+3 \mathrm{e}^{-\mathrm{x}}\right) /\left(3 \mathrm{e}^{\mathrm{x}}+4 \mathrm{e}^{-\mathrm{x}}\right)\right] \mathrm{d} \mathrm{x}=\mathrm{Ax}+\mathrm{B} \log \left|3 \mathrm{e}^{2 \mathrm{x}}+4\right|+\mathrm{c}\) then \(\mathrm{A}+\mathrm{B}=\) (a) \((11 / 24)\) (b) \((13 / 24)\) (c) \((15 / 24)\) (d) \((17 / 24)\)

\(\int\left[(x+2)^{2} /(x+4)\right] e^{x} d x=\ldots\) (a) \(\mathrm{e}^{\mathrm{x}}[\mathrm{x} /(\mathrm{x}+4)]\) (b) \(\mathrm{e}^{\mathrm{x}}[(\mathrm{x}+2) /(\mathrm{x}+4)]\) (c) \(\mathrm{e}^{\mathrm{x}}[(\mathrm{x}+2) /(\mathrm{x}+4)]\) (d) \(\left[\left(2 \mathrm{xe}^{2}\right) /(\mathrm{x}+4)\right]\)

\(\int\left[\mathrm{dx} /\left(\mathrm{e}^{\mathrm{x}}+1\right)\right]=\underline{\mathrm{c}}\) (a) \(-\log \mid\left[\left(\mathrm{e}^{\mathrm{x}}+1\right) /\left(\mathrm{e}^{\mathrm{x}}\right)\right]\) (b) \(-\log \mid\left[\left(\mathrm{e}^{\mathrm{x}}\right) /\left(\mathrm{e}^{\mathrm{x}}+1\right)\right]\) (c) \(\log \left[\left[\left(\mathrm{e}^{\mathrm{x}}+1\right) /\left(2 \mathrm{e}^{\mathrm{x}}\right)\right]\right.\) (d) \(\log \mid\left[(\mathrm{e})^{2 \mathrm{x}} /\left(\mathrm{e}^{\mathrm{x}}+1\right)\right]\)
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