Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Indefinite Integration

NTA Abhyas 2020 · Mathematics · Indefinite Integration

If the integral Ι = ∫ dx x^ 10 + x = ( x^ 9 1 + x^ μ ) + C, (where,...

If the integral I=∫dxx10+x=\lambdaln(x91+xμ)+C,Ι = ∫ \dfrac{dx}{x^{10} + x} =\lambdaln\left(\dfrac{x^{9}}{1 + x^{\mu}}\right) + C, (where, CC is the constant of integration) then the value of 1λ+μ\dfrac{1}{λ} + \mu& is equal to

Options

  1. A8181
  2. B829\dfrac{82}{9}
  3. C1818
  4. D88

Correct answer

C. 1818

Step-by-step solution

I=∫x−101+x−9dxI = ∫ \dfrac{x- 10}{1 + x^{- 9}}dx
Put 1+x−9=t1 + x^{- 9} = t
⇒−9x−10dx=dt⇒ - 9x^{- 10}dx = dt
⇒I=∫dt(−9)t⇒ Ι = ∫ \dfrac{dt}{\left(- 9\right)t}
=−19lnt+C= \dfrac{- 1}{9}ln t + C
=−19ln⁡(1+x−9)+C= \dfrac{- 1}{9}\ln\left(1 + x^{- 9}\right) + C
=19ln⁡(x91+x9)+C= \dfrac{1}{9}\ln \left(\dfrac{x^{9}}{1 + x^{9}}\right) + C
⇒λ=19,μ=9⇒ \lambda = \dfrac{1}{9}, \mu = 9
∴1λ+μ=9+9=18∴ \dfrac{1}{\lambda} + \mu = 9 + 9 = 18
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 2026Let 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 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 2024 Full Indefinite Integration list Indefinite Integration JEE Maths PYQsJEE Mathematics PYQs All NTA Abhyas JEE Main PYQs