Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
MHT CET202525 Apr 2025Morning ShiftMathematicsIndefinite IntegrationActual

d x 3 2 x+5 equals

Options

  1. A1 2 ⁻¹( x)+c , where c is the constant of integration.
  2. B1 2 ⁻¹ ( x 2 )+c, where c is the constant of integration.
  3. C1 4 ⁻¹ ( 1 2 x )+c , where c is the constant of integration.
  4. D1 4 ⁻¹( x)+ c , where c is the constant of integration.

Correct answer

C. 1 4 ⁻¹ ( 1 2 x )+c , where c is the constant of integration.

Step-by-step solution

Let I = d x 3 2x + 5 . Using the trigonometric substitution 2x = 1 - ^2 x 1 + ^2 x , we transform the integral: I = d x 3 ( 1 - ^2 x 1 + ^2 x ) + 5 = 1 + ^2 x 8 + 2 ^2 x d x . Recognizing that 1 + ^2 x = ^2 x , we have I = ^2 x d x 8 + 2 ^2 x . With the substitution t = x , where d t = ^2 x d x , the integral becomes: I = d t 8 + 2t^2 = 1 2 d t 4 + t^2 . Applying the standard antiderivative formula d u a^2 + u^2 = 1 a ( u a ) + C with a = 2 , we obtain: I = 1 4 ( t 2 ) + C . Substituting back t = x gives: I = 1 4 (

Practice Indefinite Integration on Quantrex Academy →

More from Indefinite Integration

If 5 x x -2 dx = ax + b | x -2 x |+ c , then a + b = 2025x Cos ⁻¹ ( 1-x^2 1+x^2 ) d x(x>0)= 2025d x (1+ x ) x-x^2 = 2025If x x x+1 ~d x= f (x)+ k , then f ( 4 )= 2025If x^4+1 x^2+1 d x=A x^3+B x^2+C x+D Tan ⁻¹ x+E , then A+B+C+D= 2025If x^2-x+2 x^2+x+2 d x=x- (f(x))+ 2 7 Tan ⁻¹(g(x))+c , then f (-1)+ 7 ~g (-1)= 2025(x- 3 ) (x+ 6 ) d x= 2025If a x+3 x 5 x+2 x d x= 26 29 x- k 29 |5 x+2 x|+c , then |a+k|= 2025 Full Indefinite Integration list All MHT CET PYQs