Let f-x-1-xIn -x-e-x -then its primitive with respect to x is1-2 ln x -e-x -Ce-x-2x-C1-2 ln-2 x -x -C1-2e-x-ln x -C

The correct option is C 1/2 ln^2 x x +CLet I = f(x) ∫ I dx =∫1/x(In x Ine^x)dx ∫Inx x/xdx = [ ∫1/xIn x dx ∫1/xx dx ] (Put In x =u; 1/xdx =du) = 1/2 ln^2 ...

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Quantitative Aptitude

A+B+C=N (N Fixed)

Let f(x)=1/xI...

Question

Let $f (x) = \frac{1}{x} I n (\frac{x}{e^{x}})$ then its primitive with respect to x is

\frac{1}{2} e^{x} - l n x + C

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\frac{1}{2} l n x - e^{x} + C

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\frac{1}{2} l n^{2} x - x + C

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\frac{e^{x}}{2 x} + C

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Solution

The correct option is C $\frac{1}{2} l n^{2} x - x + C$
Let I = f(x)
$\int I d x = \int \frac{1}{x} (I n x - I n e^{x}) d x$
$\int \frac{I n x - x}{x} d x = [\int \frac{1}{x} I n x d x - \int \frac{1}{x} x d x]$ (Put In x =u; $\frac{1}{x} d x = d u$ )
= $\frac{1}{2} l n^{2} x - x + C$

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Let f(x)=1xIn(xex) then its primitive with respect to x is

The correct option is C 12ln2x−x+CLet I = f(x) ∫Idx=∫1x(Inx−Inex)dx ∫Inx−xxdx=[∫1xInxdx−∫1xxdx] (Put In x =u; 1xdx=du) = 12ln2x−x+C

Let $f (x) = \frac{1}{x} I n (\frac{x}{e^{x}})$ then its primitive with respect to x is

The correct option is C $\frac{1}{2} l n^{2} x - x + C$
Let I = f(x)
$\int I d x = \int \frac{1}{x} (I n x - I n e^{x}) d x$
$\int \frac{I n x - x}{x} d x = [\int \frac{1}{x} I n x d x - \int \frac{1}{x} x d x]$ (Put In x =u; $\frac{1}{x} d x = d u$ )
= $\frac{1}{2} l n^{2} x - x + C$