Integral using Substitution (Complex Exponential Functions)

I = \int e^{3 x^{4} - 28 x^{3} + 60 x^{2} + 60 x + 4} (x^{3} - 7 x^{2} + 10 x + 5) d x

Put

u = 3 x^{4} - 28 x^{3} + 60 x^{2} + 60 x + 4

, then

d u = (12 x^{3} - 84 x^{2} + 120 x + 60) d x

Or,

\frac{1}{12} d u = (x^{3} - 7 x^{2} + 10 x + 5) d x

So,

I = \int e^{u} (\frac{1}{12}) d u

Or,

I = \frac{1}{12} \int e^{u} d u

Or,

I = \frac{1}{12} e^{u} + C

Or,

I = \frac{1}{12} e^{3 x^{4} - 28 x^{3} + 60 x^{2} + 60 x + 4} + C

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