Chapter 14, Problem 1RE

To determine

To calculate: The value of given integral ${\displaystyle \int \left(x^2+2\right)e^{x}dx}$

Answer to Problem 1RE

Solution:

The general solution is $\left(x^2-2x+4\right)e^x+C$.

Explanation of Solution

Given Information:

The provided integral ${\displaystyle \int \left(x^2+2\right)e^{x}dx}$.

Formula used:

Integration by parts

${\displaystyle \int u\cdot vdx}=u\cdot I\left(v\right)-{\displaystyle \int D\left(u\right)I\left(v\right)dx}$.

Calculation:

Consider the provided integral ${\displaystyle \int \left(x^2+2\right)e^{x}dx}$.

Integration by parts

${\displaystyle \int u\cdot vdx}=u\cdot I\left(v\right)-{\displaystyle \int D\left(u\right)I\left(v\right)dx}$.

Solve the integral by above formula as:

${\displaystyle \int \left(x^2+2\right)e^{x}dx}=\left(x^2+2\right)e^x-2x{e}^x+2e^x+C$

Put the $D$ and $I$ value in the form of table as:

	$D$	$I$
$+$	$x^2+2$	$e^x$
$-$	$2x$	$e^x$
$+$	$2$	$e^x$
$-{\displaystyle \int }$	$0$	$e^x$

Hence, the general solution is $\left(x^2-2x+4\right)e^x+C$.