Module 2: Now You Try II

Compute:

1. $LaTeX: \int_1^2 \left(x-1\right)\; dx$ $\int_1^2 \left(x-1\right)\; dx$

Solution:

▶ Show

$LaTeX: \int_1^2\left( x-1\right)\; dx=\left(\frac{x^2}{2}-x\right) \Big |_1^2=\left(\frac{2^2}{2}-2\right)-\left(\frac{1^2}{2}-1\right)$

$\int_1^2\left( x-1\right)\; dx=\left(\frac{x^2}{2}-x\right) \Big |_1^2=\left(\frac{2^2}{2}-2\right)-\left(\frac{1^2}{2}-1\right)$

2. $LaTeX: \int_0^{\frac{\pi}{4}} \sec^2 \theta\; d\theta$ $\int_0^{\frac{\pi}{4}} \sec^2 \theta\; d\theta$

Solution:

▶ Show

$LaTeX: \int_0^{\frac{\pi}{4}} \sec^2 \theta\; d\theta=\left(\tan \theta\right) \Big |_0^{\frac{\pi}{4}}=\tan(\frac{\pi}{4})-\tan(0)=1.$

$\int_0^{\frac{\pi}{4}} \sec^2 \theta\; d\theta=\left(\tan \theta\right) \Big |_0^{\frac{\pi}{4}}=\tan(\frac{\pi}{4})-\tan(0)=1.$