Evaluate integral of 1/(x^2)*cos(1/x) with respect to x

$\int \frac{1}{x^{2}} \cdot \cos (\frac{1}{x}) d x$

Let $u = \frac{1}{x}$ . Then $d u = - \frac{1}{x^{2}} d x$ , so $- x^{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Let $u = \frac{1}{x}$ . Find $\frac{d u}{d x}$ .

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Differentiate $\frac{1}{x}$ .

$\frac{d}{d x} [\frac{1}{x}]$

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$\frac{d}{d x} [x^{- 1}]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 1$ .

$- x^{- 2}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{x^{2}}$

Rewrite the problem using $u$ and $d u$ .

$\int - \cos (u) d u$

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int \cos (u) d u$

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$- (\sin (u) + C)$

Simplify.

$- \sin (u) + C$

Replace all occurrences of $u$ with $\frac{1}{x}$ .

$- \sin (\frac{1}{x}) + C$

Evaluate integral of 1/(x^2)*cos(1/x) with respect to x

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