The value of ∫ e^ x x (1 + e^ 2 x ) dx is equal to (where, C is the constant of integration)

Correct answer: C. 2 ^ -1 (e^ x ) + C Let,& I = ∫ e^ x x (1 + e^ 2 x ) dx Substituting e^ x = t e^ x 2 x dx = dt I = 2∫ dt 1 + t^ 2 = 2( ^ - 1 t) + C I = 2 ^ - 1 (e^ x ) + C

NTA Abhyas 2020

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

NTA Abhyas 2020 · Mathematics · Indefinite Integration

The value of ∫ e^ x x (1 + e^ 2 x ) dx is equal to (where, C is the...

The value of

∫ \dfrac{e^{\sqrt{x}}}{\sqrt{x}\left(1 + e^{2\sqrt{x}}\right)}

is equal to (where,

C is the constant of integration)

Options

A $\tan^{-1} \left(2e^{x}\right) + C$
B $\ln \left(\dfrac{1 + e^{\sqrt{x}}}{1 - e^{\sqrt{x}}}\right) + C$
C $2\tan^{-1} \left(e^{\sqrt{x}}\right) + C$
D $\left(\tan^{- 1} x\right)e^{\sqrt{x}} + C$

Correct answer

C. $2\tan^{-1} \left(e^{\sqrt{x}}\right) + C$

Step-by-step solution

Let,&

I = ∫ \dfrac{e^{\sqrt{x}}}{\sqrt{x}\left(1 + e^{2\sqrt{x}}\right)}dx

Substituting

e^{\sqrt{x}} = t

\dfrac{e^{\sqrt{x}}}{2\sqrt{x}}dx = dt

I = 2∫ \dfrac{dt}{1 + t^{2}}

= 2\left(\tan^{- 1} t\right) + C

I = 2\tan^{- 1} \left(e^{\sqrt{x}}\right) + C

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