Integral using Substitution (Polynomials and Rationals)

I = \int {(-100 x^{3} + 75 x^{2} - 30 x - 6)}^{3} (10 x^{2} - 5 x + 1) d x

Put u = -100 x^{3} + 75 x^{2} - 30 x - 6, then

d u = (-300 x^{2} + 150 x - 30) d x

or, (\frac{-1}{30}) d u = (10 x^{2} - 5 x + 1) d x

I = \int {(u)}^{3} (\frac{-1}{30}) d u

I = \frac{-1}{30} \int {(u)}^{3} d u

I = \frac{-1}{30} \frac{{(u)}^{4}}{4} + C

I = \frac{- {(-100 x^{3} + 75 x^{2} - 30 x - 6)}^{4}}{120} + C

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Copyright © Dayal D. Purohit, Ph.D.(Mathematics)