Q.272. Tricky Integration.

Question 272.
Evaluate: ∫ (x^27) * [(1 + x + x²)^6] * (6x² + 5x + 4)dx.

Answer 272.
∫ (x^27) * [(1 + x + x²)^6] * (6x² + 5x + 4)dx
= ∫(x^4)^6 * x^3 * [(1 + x + x²)^6] * (6x² + 5x + 4)dx
= ∫ [(x^4 + x^5 + x^6)^6] * (6x^5 + 5x^4 + 4x^3) dx
Let x^6 + x^5 + x^4 = t
=> (6x^5 + 5x^4 + 4x^3) dx = dt
=> Integral
= ∫t^6 dt
= t^7/7 + c
= (x^6 + x^5 + x^4)^7 / 7 + c
= x^28 * (1/7) (x^2 + x + 1)^7 + c.

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