First slide

Methods of integration

Question

$If I_{n} = \int x^{n} e^{α x} d x for n \geq 1, then α I_{n} + n I_{n - 1} isequal to$

Moderate

A

$x^{n} e^{α x} + C$
B

$x^{n} + C$
C

$e^{α x} + C$
D

$x^{n} + e^{α x} + C$

Solution

$I_{n} = \int x^{n} e^{α x} d x$
$\begin{array}{l} = x^{n} \frac{e^{α x}}{α} - \int n x^{n - 1} \frac{e^{α x}}{α} d x \\ Using \int f g d x = f \int g d x - \int (f^{1} \int g d x)) d x \\ I_{n} = \frac{x^{n}}{α} e^{α x} - \frac{n}{α} \int x^{n - 1} e^{α x} d x \\ α I_{n} = x^{n} e^{α x} - n I_{n - 1} \\ α I_{n} + n I_{n - 1} = x^{n} e^{α x} + C \end{array}$

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