If $f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } \,\,{n^2}({x^{1/n}}\,\, - \,\,{x^{1/\left( {n + 1} \right)}}$, $x > 0,\,\;{\rm{then}}\,\,\smallint x\;f\left( x \right)\,\,dx$ is equal to

${x^2}\,\,\log \,\,x\,\, - \,\,\frac{1}{2}\,\,{x^2} + \,\,c$

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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Analyzing +4 | -1

f\left( x \right) = \lim\limits_{n \to \infty } \,\,{n^2}({x^{1/n}}\,\, - \,\,{x^{1/\left( {n + 1} \right)}}

x > 0,\,\;\mathrm{then}\,\,\int x\;f\left( x \right)\,\,dx

is equal to

View answer and explanation

Correct option: D

None of these

Option B (

\frac{\mathrm{\pi }}{4}

) is correct because it is the result of the definite integral after the trigonometric substitution and simplification.

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