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If I = ∫ x 3 - 1 x 5 + x 4 + x + 1 d x = 1 4 ln ⁡ f x - ln ⁡ g x + c (where, c is the constant of integration) and f 0 = g 0 = 1 , then the value of f 1 ⋅ g 1 is equal to

Correct answer

4

Step-by-step solution

The given integral is I = ∫ x 3 - 1 x + 1 x 4 + 1 d x = ∫ x 3 + x 4 - x 4 - 1 x + 1 x 4 + 1 d x = ∫ x 3 x + 1 - x 4 + 1 x + 1 x 4 + 1 d x = ∫ x 3 x 4 + 1 d x - ∫ d x x + 1 = 1 4 ∫ 4 x 3 d x x 4 + 1 - ∫ d x x + 1 = 1 4 ln ⁡ x 4 + 1 - ln ⁡ 1 + x + c Hence, f x = x 4 + 1 ,   g x = x + 1 ∴ f 1 · g 1 = 2 × 2 = 4

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