# What is the net area between f(x) = cscx -cosxsinx and the x-axis over x in [pi/6, (5pi)/8 ]?

Feb 20, 2018

${\int}_{\frac{\pi}{6}}^{\frac{5 \pi}{8}} \csc x - \cos x \sin x \mathrm{dx} \approx 1.4181$

#### Explanation:

We want to find $I = {\int}_{\frac{\pi}{6}}^{\frac{5 \pi}{8}} \csc x - \cos x \sin x \mathrm{dx}$.

We do this by integrating term-by-term and then applying the limits.

$\int \csc x \mathrm{dx} = - \ln \left(\csc x + \cot x\right)$

This integral is well known

$\int - \cos x \sin x \mathrm{dx}$

Let $u = \cos x$ and $\mathrm{du} = - \sin x \mathrm{dx}$

Then $\int - \cos x \sin x \mathrm{dx} = \int u \mathrm{du} = \frac{1}{2} {u}^{2} = \frac{1}{2} {\cos}^{2} x$

So $I = {\left[\frac{1}{2} {\cos}^{2} x - \ln \left(\csc x + \cot x\right)\right]}_{\frac{\pi}{6}}^{\frac{5 \pi}{8}} \approx 1.4184$
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