If $f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {x,\,\;\;for\;\,\,x\,\, < 1}\\ {x - 1,\;\,\,for\;\,\,x\,\, \ge 1} \end{array}} \right.$, then $\mathop \smallint \limits_0^2 \,\,{x^2}\,\,f\left( x \right)dx$ is equal to

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

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Application +4 | -1

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {x,\,\;\;for\;\,\,x\,\, < 1}\\ {x - 1,\;\,\,for\;\,\,x\,\, \ge 1} \end{array}} \right.

, then

\int \limits_0^2 \,\,{x^2}\,\,f\left( x \right)dx

is equal to

View answer and explanation

Correct option: C

\frac{5}{3}

Option C (

\frac{5}{3}

) is correct because it is the result of correctly evaluating the definite integral by splitting it into two integrals based on the piecewise definition of f(x).

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\int \nolimits_0^\infty \,\,\log \,\,\left( {x + \frac{1}{x}} \right)\,\,\frac{{dx}}{{1 + {x^2}}}

is equal to If

f\left( x \right)

is differentiable and

\int \nolimits_0^{{t^2}} \,\,x\;\,f\left( x \right)\,\,dx = \frac{2}{5}\,\,{t^5}

, then

f\left( {\frac{4}{{25}}} \right)

equals The value of

\int\limits_{{0}}^{11}[x]^{3} . \mathrm{dx}

, where

[*]

denotes the greatest integer function, is The integral

\int\limits_{2}^4 \frac{\log x^2}{\log x^2+\log \left(36-12 x+x^2\right)} d x

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\int\limits_{{-2}}^0\left(x^3+3 x^2+3 x+3+(x+1) \cos (x+1)\right) d x

equals If

I\,\,\left( {m,\,\,n} \right) = \int \nolimits_0^1 \,\,{t^m}\,\,{\left( {1 + t} \right)^n}\,\,dt

,then the expression for

I\left( {m,\,\,n} \right)

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I\,\,\left( {m + 1,\,\,n + 1} \right)