Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202623 JanuaryMorning Shift
Exam
JEE Main
Year
2026
Date
23 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Indefinite Integration

JEE Main 2026 · 23 January · Morning Shift · Mathematics · Indefinite Integration

Let f(x)=∫ (2-x^ 2 ) · e^ x ( 1+x )(1-x)^ 3 / 2 ~d x . If f(0)=0 ,...

Let f(x)=∫(2−x2)⋅ex(1+x)(1−x)3/2 dxf(x)=\int \frac{\left(2-x^{2}\right) \cdot \mathrm{e}^{x}}{(\sqrt{1+x})(1-x)^{3 / 2}} \mathrm{~d} x. If f(0)=0f(0)=0, then f(12)f\left(\frac{1}{2}\right) is equal to:

Options

  1. A3e−1\sqrt{3 \mathrm{e}}-1
  2. B3e+1\sqrt{3 \mathrm{e}}+1
  3. C2e+1\sqrt{2 \mathrm{e}}+1
  4. D2e−1\sqrt{2 \mathrm{e}}-1

Correct answer

A. 3e−1\sqrt{3 \mathrm{e}}-1

Step-by-step solution

Write 2−x2=(1−x)(1+x)+12 - x^2 = (1-x)(1+x) + 1, so the integrand splits as ex(1+x1−x+11+x(1−x)3/2)e^x\left(\sqrt{\frac{1+x}{1-x}} + \frac{1}{\sqrt{1+x}(1-x)^{3/2}}\right).
Let g(x)=1+x1−xg(x) = \sqrt{\frac{1+x}{1-x}}. Then g′(x)=11+x(1−x)3/2g'(x) = \frac{1}{\sqrt{1+x}(1-x)^{3/2}}.
Using ∫ex(g+g′)dx=exg+C\int e^x(g + g')dx = e^x g + C: f(x)=ex1+x1−x+Cf(x) = e^x\sqrt{\frac{1+x}{1-x}} + C.
f(0)=1+C=0⇒C=−1f(0) = 1 + C = 0 \Rightarrow C = -1.
f(1/2)=e1/23/21/2−1=e⋅3−1=3e−1f(1/2) = e^{1/2}\sqrt{\frac{3/2}{1/2}} - 1 = \sqrt{e}\cdot\sqrt{3} - 1 = \sqrt{3e} - 1.
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 2026Let f(x)=∫ d x x^ ((2)/(3)) +2 x^ ((1)/(2)) be such that f(0)=-26+24 _ e (2) . If… JEE Main 2026Let 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 2026Let 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 2026If ∫( x)^ (-11)/(2) ( x)^ (-5)/(2) d x= - p_ 1 q_ 1 ( x)^ (9)/(2) - p_ 2 q_ 2 ( x)^… JEE Main 2026If aligned ∫((1)/(x)+(1)/(x^3)) & ( [23] 3 x^ -24 +x^ -26 ) d x & =-(α)/(3(α+1))(3… JEE Main 2025If ∫ ( 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 2025Let f(x)=∫ x^3 3-x^2 d x . If 5 f( 2 )=-4 , then f(1) is equal to JEE Main 2025If f(x)=∫ 1 x^ 1 / 4 (1+x^ 1 / 4 ) d x, f(0)=-6 , then f(1) is equal to : JEE Main 2025If ∫ 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 2025Let ∫ x^3 x ~d x=g(x)+C , where C is the constant of integration. If 8(g((π)/(2))+g^… JEE Main 2025Let I(x)=∫ d x (x-11)^ (11)/(13) (x+15)^ (15)/(13) . If I(37)-I(24)=(1)/(4)( 1 ~b^… JEE Main 2025If ∫ 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 2025Let ∫ (2- x)/(3+ x) ~d x=(1)/(2)(α x+ _ e |β x+γ x|)+C , where C is the constant of… JEE Main 2024If ∫ 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 2024Let I(x)=∫ (6)/( ^2 x(1- x)^2) d x . If I(0)=3 , then I((π)/(12)) is equal to JEE Main 2024 Full Indefinite Integration list Indefinite Integration JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs