Find the value of: e x ( 1 x 4 - 4 x 5 ) ? d x

Question

Find the value of:

$\int e^{x} (\frac{1}{x^{4}} - \frac{4}{x^{5}}) d x$

Smart Achievers · Accepted Answer

(1)

Ans. $e^{x} \frac{1}{x^{4}} + C$

Explanation :

The given integral is,

$I = \int e^{x} (\frac{1}{x^{4}} - \frac{4}{x^{5}}) d x$

Let, $f (x) = \frac{1}{x^{4}}$

Differentiating $f (x)$ ,

$f' (x) = - \frac{4}{x^{5}}$

Substituting $f (x)$ for $\frac{1}{x^{4}}$ and $f' (x)$ for $- \frac{4}{x^{5}}$ in the given integral, we get

$I = \int e^{x} (\frac{1}{x^{4}} - \frac{- 4}{x^{5}}) d x$

$= \int e^{x} {f (x) + f' (x)} d x$

We know that

$\int e^{x} {f (x) + f' (x)} d x = e^{x} f (x) + C$

$∴$ $I = e^{x} \frac{1}{x^{4}} + C$

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