If ∫ d x x + x 3 = a x + b x 3 + c x 6 + d ln ( x 6 + 1 ) + e , e being an arbitrary constant, then the value of 20 a + b + c + d is

Question

If ∫ d x x + x 3 = a x + b x 3 + c x 6 + d ln ( x 6 + 1 ) + e , e being an arbitrary constant, then the value of 20 a + b + c + d is,

MARKS App · Accepted Answer

Let x = u 6 , d x = 6 u 5 d u ∫ d x x + x 3 = ∫ 6 u 5 d u u 3 + u 2 = 6 ∫ u 3 u + 1 d u = 6 ∫ u 2 - u + 1 - 1 u + 1 d u = 2 u 3 - 3 u 2 + 6 u - 6 ln u + 1 +e = 2 x - 3 x 3 + 6 x 6 - 6 ln x 6 + 1 +e ∴ a = 2 , b = - 3 , c = 6 , d = - 6 ∴ 20 a + b + c + d = 37

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