Evaluate the integral.

{eq}\int_{0}^{\frac{\pi}{4}} \frac{7 + 9 \cos^2 \theta}{\cos^2 \theta} \, \mathrm{d} \theta {/eq}

Question:

Definite Integral:

The definite integral uses the fundamental theorem of calculus to evaluate the integral. The definite integral comes with the two boundaries with the integral. For example, {eq}\displaystyle \int_a^b v(r) dr = V(b)-V(a). {/eq}

Certain properties are useful in solving problems:

1. The sum rule: {eq}\displaystyle \int f\left(t\right)\pm g\left(t\right)dt=\int f\left(t\right)dt\pm \int g\left(t\right)dt. {/eq}

2. Remove the constant from the integration: {eq}\displaystyle \int a\cdot f\left(t\right)dt=a\cdot \int f\left(t\right)dt. {/eq}

3. Common integration: {eq}\displaystyle \int \frac{1}{\cos ^2\left(\theta\right)}d\theta=\tan \left(\theta\right). {/eq}

4. Value of {eq}\displaystyle \tan \left(0\right)=0. {/eq}

5. Value of {eq}\displaystyle \tan \left(\frac{\pi }{4}\right)=1. {/eq}

Answer and Explanation: 1