# Calculate the following antiderivative. Show your work. \int \left( x+ \frac {1}{x}-\sin (x)...

## Question:

Calculate the following antiderivative. Show your work.

{eq}\int \left( x+ \frac {1}{x}-\sin (x) \right ) dx {/eq}

## Answer and Explanation:

We are given:

{eq}\displaystyle \int \left( x+ \frac {1}{x}-\sin (x) \right ) dx {/eq}

Apply integral sum rule:

{eq}=\displaystyle \int x\ dx + \int \frac{1} {x} \ dx- \int \sin(x) \ dx {/eq}

Apply common integrals:

{eq}=\displaystyle \dfrac{x^{1+1}}{1+1} +\ln x +\cos(x) + C {/eq}

{eq}=\displaystyle \dfrac{x^{2}}{2} +\ln x +\cos(x) + C {/eq}

Therefore the solution is:

{eq}\displaystyle{\boxed{ \int \left( x+ \frac {1}{x}-\sin (x) \right ) dx=\dfrac{x^{2}}{2} +\ln x +\cos(x) + C . }} {/eq}

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