Problems on Techniques of Integration

Use the Integration by Parts. Set

$$\left\{\begin{array}{lll} u &=& x^2\\ dv &=& e^x dx\;. \end{array}\right.$$

Then

$$\left\{\begin{array}{lll} du &=&2 x dx\\ v &=& e^x\;. \end{array}\right.$$

So

$$\int x^2 e^x dx = x^2 e^x - \int 2 x e^x dx = x^2 e^x - 2 \int x e^x dx\;.$$

In order to integrate the function $x e^x$, we will need to do another integration by parts. Set

$$\left\{\begin{array}{lll} u &=&x\\ dv &=& e^{x}dx\;. \end{array}\right.$$

Then

$$\left\{\begin{array}{lll} du &=&dx\\ v &=& e^{x}\;. \end{array}\right.$$

So

$$\int x e^{x} dx = x e^{x} - \int e^{x} dx$$

or

$$\int x e^{x} dx = x e^x - e^{x} \;.$$

Hence

$$\int x^2 e^x dx = x^2 e^x - 2 x e^x + 2 e^{x} + C = \Big(x^2 - 2 x + 2\Big)e^x + C\;.$$

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