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Most Important Selected Qs for JEE AdvancedMathematicsDefinite Integration

Let f:(0,1) R be a function defined by f(x)=10 x^ x+ x . If I= ₀^1 f(x) d x , then find the value of [I] . [Note: Where [k] denotes the greatest integer function less than or equal to k .]

Correct answer

4

Step-by-step solution

x+ x= 2 (x- 4 ) [1, 2 ] x [0,1] Hence, as x [0,1] 10 x^ 2 f(x) 10 x With equality holding if and only if x=0 or 1 As equality holds only for finitely many points, the inequalities become on strict integrating on all sides. Hence, 4 < 10( 2 -1) < ₀^1 f(x) d x < 5 [ ₀^1 f(x) d x ]=4

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