Let ∫ e ^ x ⋅ x ^ 2 dx = f(x) e ^ x + C (where, C is the constant of integration). The range of f(x) as x ∈ R is [a, ∞) . The value of (a)/(4) is

Correct answer: 0.25 As we know, ∫ e ^ x (f(x) + f ^ ' (x))dx = e ^ x ⋅ f(x) + C, Thus, ∫ e ^ x ⋅ x ^ 2 dx =∫ e ^ x ( x ^ 2 + 2x - 2x - 2 + 2)dx = ∫ [ e ^ x ( x ^ 2 - 2x + 2) + e ^ x (2x - 2)]dx = e ^ x ( x ^ 2 - 2x + 2) + C i.e. f(x) = x ^ 2 - 2x + 2 = (x - 1) ^ 2 + 1 Hence, range is [1, ∞) ⇒ a = 1 ∴ (a)/(4) = 0.25

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Indefinite Integration

NTA Abhyas 2020 · Mathematics · Indefinite Integration

Let ∫ e ^ x ⋅ x ^ 2 dx = f(x) e ^ x + C (where, C is the constant...

Let

∫ {e}^{x} ⋅ {x}^{2}dx = f\left(x\right){e}^{x} + C

(where,

C is the constant of integration). The range of

f\left(x\right)

as

x \in R

is \left[a, \infty\right) .

The value of

\dfrac{a}{4}

Correct answer

0.25

Step-by-step solution

As we know,

∫ {e}^{x}\left(f\left(x\right) + {f}^{'}\left(x\right)\right)dx = {e}^{x} ⋅ f\left(x\right) + C,

Thus,

∫ {e}^{x} ⋅ {x}^{2}dx =∫ {e}^{x}\left({x}^{2} + 2x - 2x - 2 + 2\right)dx

= ∫ \left[{e}^{x}\left({x}^{2} - 2x + 2\right) + {e}^{x}\left(2x - 2\right)\right]dx

= {e}^{x}\left({x}^{2} - 2x + 2\right) + C

i.e.

f\left(x\right) = {x}^{2} - 2x + 2 = {\left(x - 1\right)}^{2} + 1

Hence, range is \left[1, \infty\right) ⇒ a = 1

<br />

∴ \dfrac{a}{4} = 0.25

Practice Indefinite Integration on Quantrex Academy →

More from Indefinite Integration

Let f(x) = ∫ ((16x + 24)/(x^2 + 2x - 15)) dx . If f(4) = 14 _e(3) and f(7) = _e(2^ α · 3^… JEE Main 2026 Let f(x)=∫ d x x^ ((2)/(3)) +2 x^ ((1)/(2)) be such that f(0)=-26+24 _ e (2) . If… JEE Main 2026 Let f(t)=∫( 1- ( _ e t) 1- ( _ e t) ) d t, t>1 . If f(e^ π / 2 )=-e^ π / 2 and f(e^ π / 4… JEE Main 2026 Let I(x)=∫ 3 d x (4 x+6)( 4 x^ 2 +8 x+3 ) and I(0)= 3 4 +20 . If I((1)/(2))= a 2 b +c… JEE Main 2026 Let f(x)=∫ (2-x^ 2 ) · e^ x ( 1+x )(1-x)^ 3 / 2 ~d x . If f(0)=0 , then f((1)/(2)) is… JEE Main 2026 If ∫( x)^ (-11)/(2) ( x)^ (-5)/(2) d x= - p_ 1 q_ 1 ( x)^ (9)/(2) - p_ 2 q_ 2 ( x)^… JEE Main 2026 If aligned ∫((1)/(x)+(1)/(x^3)) & ( [23] 3 x^ -24 +x^ -26 ) d x & =-(α)/(3(α+1))(3… JEE Main 2025 If ∫ ( 1+x^2 +x)^ 10 ( 1+x^2 -x)^9 d x= (1)/(m)(( 1+x^2 +x)^n(n 1+x^2 -x))+C where C is… JEE Main 2025 Let f(x)=∫ x^3 3-x^2 d x . If 5 f( 2 )=-4 , then f(1) is equal to JEE Main 2025 If f(x)=∫ 1 x^ 1 / 4 (1+x^ 1 / 4 ) d x, f(0)=-6 , then f(1) is equal to : JEE Main 2025 If ∫ 2 x^2+5 x+9 x^2+x+1 ~d x=x x^2+x+1 +α x^2+x+1 +β _e|x+(1)/(2)+ x^2+x+1 |+C , where C… JEE Main 2025 Let ∫ x^3 x ~d x=g(x)+C , where C is the constant of integration. If 8(g((π)/(2))+g^… JEE Main 2025 Let I(x)=∫ d x (x-11)^ (11)/(13) (x+15)^ (15)/(13) . If I(37)-I(24)=(1)/(4)( 1 ~b^… JEE Main 2025 If ∫ e^x( x ^ -1 x 1-x^2 + ^ -1 x (1-x^2)^ 3 / 2 +(x)/(1-x^2)) d x=g(x)+C , where C is… JEE Main 2025 Let ∫ (2- x)/(3+ x) ~d x=(1)/(2)(α x+ _ e |β x+γ x|)+C , where C is the constant of… JEE Main 2024 If ∫ 1 [5] (x-1)^4(x+3)^6 ~d x=A((α x-1)/(β x+3))^B+C , where C is the constant of… JEE Main 2024 Full Indefinite Integration list Indefinite Integration JEE Maths PYQs JEE Mathematics PYQs All NTA Abhyas JEE Main PYQs