# How do you find the definite integral of int (1-costheta)/(theta-sintheta) from [1,2]?

May 10, 2017

$\ln \left(2 - \sin \left(2\right)\right) - \ln \left(1 - \sin \left(1\right)\right) \approx 1.9286$

#### Explanation:

Step 1. Use $u$-substitution
Let $u = \theta - \sin \left(\theta\right)$
$\frac{\mathrm{du}}{d \theta} = \left(1 - \cos \left(\theta\right)\right) d \theta$

Step 2. Use this new information to create new limits of integration
$\theta = 1$ becomes $u = 1 - \sin \left(1\right)$
$\theta = 2$ becomes $u = 2 - \sin \left(2\right)$

Step 3. Plug these substitutions into the integral
${\int}_{1 - \sin \left(1\right)}^{2 - \sin \left(2\right)} \frac{1}{u} \mathrm{du}$

Step 4. Integrate equation with $u$
${\int}_{1 - \sin \left(1\right)}^{2 - \sin \left(2\right)} \frac{1}{u} \mathrm{du} = {\left[\ln \left(u\right)\right]}_{1 - \sin \left(1\right)}^{2 - \sin \left(2\right)}$
$= \ln \left(2 - \sin \left(2\right)\right) - \ln \left(1 - \sin \left(1\right)\right) \approx 1.9286$