Definite IntegrationmediumFree

∫ e^{5x⁶}(x−α) dx = 0 implies range of α | JEE

JEE Maths question with a full step-by-step solution.

Question

\displaystyle\int_{0}^{1}e^{5x^{6}}(x-\alpha)\,dx=0

, then

1<\alpha<2

\alpha<0

0<\alpha<1

correct

\alpha=0

Solution

Step 1: Interpret the condition geometrically. The integral being zero means the signed area enclosed by

h(x)=e^{5x^{6}}(x-\alpha)

and the

x

-axis over

[0,1]

cancels out. For this cancellation,

h

must change sign somewhere inside

(0,1)

. Step 2: Examine the sign of

h

. The factor

e^{5x^{6}}>0

for all

x

, so the sign of

h(x)

is the sign of

(x-\alpha)

. Step 3: For

(x-\alpha)

to change sign within

(0,1)

, the root

x=\alpha

must lie strictly inside the interval:

0<\alpha<1.

Step 4: If

\alpha\le 0

then

x-\alpha>0

throughout

(0,1)

and the integral is positive; if

\alpha\ge 1

then

x-\alpha<0

throughout and it is negative. Only

0<\alpha<1

allows a zero value. Correct answer: (3)

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