The value of \(\rm \displaystyle\int x^2 e^{ax} dx\) is

Concept:

The product of function can be integrated by the method of "Integration by parts".

By the method of integration by parts we have,

\(\smallint f\left( x \right)g\left( x \right)dx\; = \;f\left( x \right)\smallint g\left( x \right)dx - \smallint \left[ {f'\left( x \right)\smallint g\left( x \right)dx} \right]dx\)

Where f is the first function and g is the second function.

Which is chosen based on the order for the selection of the first function: ILATE (Inverse, Logarithmic, Algebraic, Trigonometric, Exponent)

Calculation:

From the above formula,

\(\int x^2e^{ax}dx=\frac{x^2e^{ax}}{a}-\int \frac{2xe^{ax}}{a}dx\)

⇒ \(x^2e^{ax}-\frac{2}{a}[\frac{xe^{ax}}{a}-\frac{e^{ax}}{a^2}]\)

⇒ \(x^2e^{ax}-\frac{2}{a^2}xe^{ax}+\frac{2}{a^3}e^{ax}\)

Hence, \(\rm \displaystyle\int x^2 e^{ax} dx\) = \(\rm \frac{e^{ax}}{a^3} (a^2 x^2 -2ax + 2)\)