Integral using Substitution (Complex Exponential Functions)

I = \int e^{x^{4} + 4 x^{3} - 2 x^{2} - 16 x - 4} (x^{3} + 3 x^{2} - x - 4) d x

Put

u = x^{4} + 4 x^{3} - 2 x^{2} - 16 x - 4

, then

d u = (4 x^{3} + 12 x^{2} - 4 x - 16) d x

Or,

\frac{1}{4} d u = (x^{3} + 3 x^{2} - x - 4) d x

So,

I = \int e^{u} (\frac{1}{4}) d u

Or,

I = \frac{1}{4} \int e^{u} d u

Or,

I = \frac{1}{4} e^{u} + C

Or,

I = \frac{1}{4} e^{x^{4} + 4 x^{3} - 2 x^{2} - 16 x - 4} + C

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