Integral using Substitution (Complex Exponential Functions)

I = \int e^{4 x^{3} + 30 x^{2} + 36 x + 3} (x^{2} + 5 x + 3) d x

Put

u = 4 x^{3} + 30 x^{2} + 36 x + 3

, then

d u = (12 x^{2} + 60 x + 36) d x

Or,

\frac{1}{12} d u = (x^{2} + 5 x + 3) 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^{4 x^{3} + 30 x^{2} + 36 x + 3} + C

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