Question:
$\displaystyle\int_{1/2}^{2}|\log_{10}\,x|dx= $
$\displaystyle\int_{1/2}^{2}|\log_{10}\,x|dx= $
Updated On: Jul 6, 2022
- $\log_{10}\left(\frac{8}{e}\right) $
- $\frac{1}{2}\log_{10}\left(\frac{8}{e}\right) $
- $log_{10}\left(\frac{2}{e}\right) $
- none of these.
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The Correct Option is B
Solution and Explanation
$\int log \, x \, dx=x \,log x-x$
[By Using integration by parts]
$log x is -ve. for x, < 1$ and $+ve$ for $x > 1 $.
Also $log_{10} \,x= \frac{log\,x}{log\,10}$
$\therefore$ given integral
$=\frac{1}{log\,10}\left[\int\limits_{1 /2}^{1}-log\,x\,dx+\int\limits_{1}^{2} log\,x\,dx\right]$
$=-\frac{1}{log\,10} \left[x\,log\,x-x\right]_{1 /2}^{1}+\frac{1}{log\,10} \left[x\,log\,x-x\right]_{1}^{2}$
$=-\frac{1}{log\,10}\left[-1-\frac{1}{2}log\frac{1}{2}+\frac{1}{2}\right]+\frac{1}{log\,10} \left[2\,log\,2-2+1\right]$
$=\frac{1}{log\,10}\left[\frac{1}{2}-\frac{1}{2}log\,2\right]+\frac{2\,log\,2-1}{log\,10}$
$=\frac{4\,log\,2-2+1-log\,2}{2\,log\,10}=\frac{3\,log\,2-1}{2\,log\,10}$
$=\frac{1}{2} \frac{log\,8-log\,e}{log\,10}=\frac{1}{2} \frac{log\left(\frac{8}{e}\right)}{log\,10}$
$=\frac{1}{2}log_{10} \left(\frac{8}{e}\right)$
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Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.