Answer to Question #104614 in Calculus for Hashim Hussain Shah

Question

The complex function can be modelled by equation y=3x^2 x e^x3 Find the indefinite integral of function using substitution method

Answer 1

The function can be modelled by the equation given by

$y = 3x^2 e^{x^3}$

The Indefinite integral of this equation is:

$\int y \, dx$

$= \int 3x^2 e^{x^3} \, dx$

Let us assume that:

$x^3 = u \\ \Rightarrow 3x^2 \, dx = du$

Substituting $x^3 = u$ into the integration and integrating with respect to $du$ , we have:

$= \int e^u \, du \\$

$= e^u + C \hspace{1 cm}$ [ $\because \int e^x \, dx = e^x + C \,\,$ (where C is a constant of integration) ]

Undo substitution and we have:

$= e^{x^3} + C \hspace{ 1 cm} \left[ \because u = x^3 \right]$