Integral using Substitution (Complex Exponential Functions)

I = \int e^{2 x^{3} + 27 x^{2} + 42 x - 2} (x^{2} + 9 x + 7) d x

Put

u = 2 x^{3} + 27 x^{2} + 42 x - 2

, then

d u = (6 x^{2} + 54 x + 42) d x

Or,

\frac{1}{6} d u = (x^{2} + 9 x + 7) d x

So,

I = \int e^{u} (\frac{1}{6}) d u

Or,

I = \frac{1}{6} \int e^{u} d u

Or,

I = \frac{1}{6} e^{u} + C

Or,

I = \frac{1}{6} e^{2 x^{3} + 27 x^{2} + 42 x - 2} + C

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