Step by step solution to "Use the reduction formulas in a table of..."

shadsiei

Answered question

2020-11-09

Use the reduction formulas in a table of integrals to evaluate the following integrals.

\int x^{3} e^{2 x} d x

izboknil3

Skilled2020-11-10Added 99 answers

Given I = \int x^{3} e^{2 x} d x

For evaluating given integral, we use integral by parts theorem

According to integral by parts theorem

\int f (x) g^{'} (x) d x = f (x) \int g^{'} (x) d x - [\int f^{'} (x) \int g^{'} (x) d x) d x]

Here, f (x) = x^{3}, g^{'} (x) = e^{2 x}

So, be using equation

I = \int x^{3} e^{2 x} d x

= x^{3} \int e^{2 x} d x - [\int (\frac{d}{d x} (x^{3}) \int e^{2 x} d x) d x]

(∵ \int e^{k x} d x = \frac{e^{k x}}{k} + c, \frac{d}{d x} (x^{n}) = (n x^{n - 1})

= x^{3} (\frac{e^{2 x}}{2}) - [\int (3 x^{2}) (\frac{e^{2 x}}{2} d x]

= \frac{x^{3} e^{2 x}}{2} - \frac{3}{2} [\int x^{2} e^{2 x} d x]

= \frac{x^{3} e^{2 x}}{2} - \frac{3}{2} [x^{2} \int e^{2 x} d x - [\int [\frac{d}{d x} (x^{2}) \int e^{2 x} d x] d x]]

= \frac{x^{3} e^{2 x}}{2} - \frac{3}{2} [x^{2} (\frac{e^{2 x}}{2}) - [\int (2 x) \frac{e^{2 x}}{2} d x]]

= \frac{x^{3} e^{2 x}}{2} - \frac{3 x^{2} e^{2 x}}{4} + \frac{3}{2} \int x e^{2 x} d x

= \frac{x^{3} e^{2 x}}{2} - \frac{3 x^{2} e^{2 x}}{4} + \frac{3}{2} [x \int e^{2 x} d x - [\int (\frac{d}{d x} (x) \int e^{2 x} d x) d x]]

= \frac{x^{3} e^{2 x}}{2} - \frac{3 x^{2} e^{2 x}}{4} + \frac{3}{2} [x (\frac{e^{2 x}}{2} - [\int (1) (\frac{e^{2 x}}{2}) d x]]

= \frac{x^{3} e^{2 x}}{2} - \frac{3 x^{2} e^{2 x}}{4} + \frac{3 x e^{2 x}}{4} - \frac{3 e^{2 x}}{8} + c

= \frac{(4 x^{3} - 6 x^{2} + 6 x - 3) e^{2 x}}{8} + c

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