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If integration ln ex^x 1 (ln(x^√x))^2 dx /(1 xlnx)lne^2 x^x = f(x) c where f(1)=0 then e^(e^f(2) -1) is equal to?
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If integration ln ex^x 1 (ln(x^√x))^2 dx /(1 xlnx)lne^2 x^x = f(x) c ...
Integration of ln(ex^x(1+(ln(x^√x))^2) dx/(1+xlnx)ln(e^2x^x) = f(x) + C


Step 1: Simplifying the integral


To simplify the integral, we can use the following properties of logarithms:

- ln(ab) = ln(a) + ln(b)
- ln(a/b) = ln(a) - ln(b)
- ln(a^b) = b ln(a)

Using these properties, we can simplify the integral as follows:

ln(ex^x(1+(ln(x^√x))^2) dx/(1+xlnx)ln(e^2x^x)

= ln(e) + ln(ex^x) + ln(1+(ln(x^√x))^2) - ln(1+xlnx) - ln(e^2x^x)

= x + ln(1+(ln(x^√x))^2) - ln(1+xlnx) - 2x^x

Step 2: Finding f(x)


To find f(x), we need to integrate the simplified expression:

f(x) = ∫(x + ln(1+(ln(x^√x))^2) - ln(1+xlnx) - 2x^x) dx

f(x) = ½ x^2 + x ln(1+(ln(x^√x))^2) - ∫ln(1+xlnx) dx - (2/ln(e^2))

Using integration by parts, we can find the integral of ln(1+xlnx) as follows:

Let u = ln(1+xlnx) and dv = dx
Then du/dx = 1/(1+xlnx) + (lnx)/sqrt(x)
And v = x - (1/2) ln(1+xlnx)

Using the formula for integration by parts, we get:

∫ln(1+xlnx) dx = x ln(1+xlnx) - ∫[(x/sqrt(x)) + (1/2)(1/(1+xlnx))] dx

= x ln(1+xlnx) - ½ x^2 - ½ ln(1+xlnx) + ∫(1/(2(1+xlnx))) dx

= x ln(1+xlnx) - ½ x^2 - ½ ln(1+xlnx) + (1/2) ln(1+xlnx) - (1/2) ln(ln(e^2))

= x ln(1+xlnx) - ½ x^2

Substituting this back into the expression for f(x), we get:

f(x) = ½ x^2 + x ln(1+(ln(x^√x))^2) - x ln(1+xlnx) + ½ x^2 + (2/ln(e^2))

= x ln((1+(ln(x^√x))^2)/(1+xlnx)) + (2/ln(e^2))

Step 3: Finding e^(e^f(2) -1)


To find e^(e^f(2) -1), we need
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Not differentiable then S 2019 won is an empty set to equals minus 2 minus 1 0 1 2 3 equals minus 2 minus 1 1 2 4 equals minus 2 2 ANS 2 solution FX is not differentiable at negative 2 minus 1 and 2 s equals minus 2 minus 1 0 1 2 qu 21 let F be a differentiable function such than FXX greater than 0 and F14 then 2019 1 exists and equals 4/7 2 exists and equals 4 3 does not exist 4 exists and equals 0 ANS 2 solutions reply equals FX solution of differential equation 22 let's denote the greatest integer less than or equal to 10 spend 2019 equals pi plus 1 11 equals 0 and 1 solution just since it does not exist You 23 2019-102 like four for one ANS 4 solution huge 24 let F equals RB differentiable at CR and F C equals 0 If GX equals FX then at X equals C G is 20191 not differentiable if FC equals 0 to differentiable if FC03 differentiable if FC equals 04 not differentiable ANS 3 solution Q25 is continuous at x equals 0 than the ordered pair PQ is equal to 2019 ANS 3 solution Q26 LED FRRB a continuously differentiable function such that F2 equals 6 and F2 equals 148 2019 1 18/24 3/12/436 ANS 1 solution Q27 2019/1424 square root 2 3/8 square root 248 A for solution Q28 2019 ANS 2 solution huge 29 if the function has to find on is continuous then K is equal to 2019 1 2 2 1 have 3 1 4 1 / square Benz FX is continuous then Q30 Let FX equals 15 minus X 10 X are then the set of all values of X at This is not differentiable is 2019-15 to 10 15 3 5 10 15 24 10 ANS 1 solution Since FX equals 15 minus 10X GX equals FFX equals 15 minus 10 minus 15 minus 10X equals 15 10X 5 then the points were function GX is non-differential or 10 X equals 0 and 10 X equals 5X equals 10 and X 10 equals plus minus 5 X equals 10 and X equals where it's denotes the greatest integer function then 2019 one F is continuous at X equals 4 ANI One solution Q32 if the function is continuous at x equals 5 then the value of A B is 2019 ANS 4 solution function is continuous at x equals 5 LHL equals RHL5 minus pi B plus 3 equals 5 minus pi A plus 1 Q3 and 1914 square root 22 square root 232 square root 244 ANS 1 solution Q34 let FRRB a differential function satisfying F3 plus F2 equals 0 then is equal to 2019 1 1 2 E 1 3 4 E2ANS 1 solution Q35 if FRR is a differentiable function and F2 equals 6 then 2019/124F222F230412F2 ANS 4 solution Using L hospital rule and Live NetSerum we get putting X = 2 2 F2 F2 = 12 F2 F2 = 6 Q36 2019/18/33 seconds 4/3 AMS One solution GX = FX + FX Then in the Interval - 2/2 G is one differentiable at all points two not continuous three not differentiable at two points four not differentiable at one point ANS 4 solution GX is non-differentiable at x equals 1 GX is not differentiable at one point Q38 let KB the set of all real values of x were the function fx = sin x minus x plus 2x minus pi cos x is not differentiable Then the set k is equal to 2019 empty set 2 pi 3 0 4 0 pi ANS 1 solution Then function f x is differentiable for l Related: JEE Main Previous year questions (2016-20): Limits, Continuity and Differentiability?

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If integration ln ex^x 1 (ln(x^√x))^2 dx /(1 xlnx)lne^2 x^x = f(x) c where f(1)=0 then e^(e^f(2) -1) is equal to?
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