Integral using Substitution (Complex Exponential Functions)

I = \int e^{4 x^{5} - 50 x^{4} + 30 x^{2} + 180 x + 5} (x^{4} - 10 x^{3} + 3 x + 9) d x

Put

u = 4 x^{5} - 50 x^{4} + 30 x^{2} + 180 x + 5

, then

d u = (20 x^{4} - 200 x^{3} + 60 x + 180) d x

Or,

\frac{1}{20} d u = (x^{4} - 10 x^{3} + 3 x + 9) d x

So,

I = \int e^{u} (\frac{1}{20}) d u

Or,

I = \frac{1}{20} \int e^{u} d u

Or,

I = \frac{1}{20} e^{u} + C

Or,

I = \frac{1}{20} e^{4 x^{5} - 50 x^{4} + 30 x^{2} + 180 x + 5} + C

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