Evaluate the following integrals: ∫ 2x3\(e^{x^2}\)dx

1.	Evaluate the following integrals: ∫ 2x3\(e^{x^2}\)dx
Answer» Let I = ∫ 2x³\(e^{x^2}\)dx Put x²= t 2xdx = dt I = \(\int t\,e^tdt\) Using integration by parts, = \(t\int e^tdt\) - \(\int\frac{d}{dt}t\int e^t\,dt\) We have, \(\int e^xdx\) = e^x = te^t - e^t + c = e^t(t - 1) + c Substitute value for t, I = \(e^{x^2}\)(x² - 1) + c

Answer»

Let I = ∫ 2x³\(e^{x^2}\)dx

Put x²= t

2xdx = dt

I = \(\int t\,e^tdt\)

Using integration by parts,

= \(t\int e^tdt\) - \(\int\frac{d}{dt}t\int e^t\,dt\)

We have,

\(\int e^xdx\) = e^x

= te^t - e^t + c

= e^t(t - 1) + c

Substitute value for t,

I = \(e^{x^2}\)(x² - 1) + c

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