If ∫ x 4 + x 3 − x 2 − 3 x + 9 x 2 + 3 d x = n m tan − 1 ( n x m ( x − α ) ( x − β ) ) + C , where α β & α , β ∈ R and m \& n are prime numbers, then (where C is constant of integration)

Question

If ∫ x 4 + x 3 − x 2 − 3 x + 9 x 2 + 3 ​ d x = n ​ m ​ tan − 1 ( n ​ x m ( x − α ) ( x − β ) ​ ) + C , where α > β & α , β ∈ R and m \& n are prime numbers, then (where C is constant of integration)

TestHub · Accepted Answer

∫ ( x − x 3 ​ ) 2 + x − x 3 ​ + 5 ( 1 + x 2 3 ​ ) d x ​ = ∫ y 2 + y + 5 d y ​ where y = x − x 3 ​ = ∫ ( y + 2 1 ​ ) 2 + ( 2 19 ​ ​ ) 2 dy ​ = 19 ​ 2 ​ tan − 1 ( 19 ​ 2 y + 1 ​ ) + C = 19 ​ 2 ​ tan − 1 ( 19 ​ x 2 ( x + 2 ) ( x − 2 3 ​ ) ​ ) + C ⇒ α = 2 3 ​ & β = − 2 m = 2 & n = 19

Mathematics - Indefinite Integration Question with Solution | TestHub

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