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AP EAMCET2010MathematicsIndefinite Integration

If f_n(x)= x ( is repeated n -times), then (x f₁(x) f₂(x) f_n(x) )⁻¹ d x is equal to

Options

  1. Af_ n+1 (x)+c
  2. Bf_ n+1 (x) n+1 +c
  3. Cn f_n(x)+c
  4. Df_n(x) n +c

Correct answer

A. f_ n+1 (x)+c

Step-by-step solution

f_n(x)= x (upto n times) f₁(x)= xf₂(x)= xf₃(x)= xf_ n-1 (x)= x (up to (n-1) times) Now, (x f₁(x) f₂(x) f_n(x) )⁻¹ d x= d x [x f₁(x) f₂(x) f_n(x) ] = [x f₁(x) f₂(x) f_ n-1 (x) ] d t [x f₁(x) f₂(x) f_ n-1 (x) ] t = d t t = t+c [ array l Put, f_n(x)=t d t d x = 1 [f_ n-1 (x) f_ n-2 (x) f₁(x) x ] d x= [x f₁(x) f₂(x) f_ n-1 (x) ] d t array ]= f_n(x)+c=f_ n+1 (x)+c Hence, (x f₁(x) f₂(x) f_n(x) )⁻¹ d x=f_ n+1 (x)+c

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