Evaluate the integral.

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


Evaluate the integral.

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

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

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We have to evaluate the integration of $$\displaystyle I = \int_{0}^{\frac{\pi}{4}} \frac{7 + 9 \cos^2 \theta}{\cos^2 \theta} \, \mathrm{d}...

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Evaluating Definite Integrals Using the Fundamental Theorem


Chapter 16 / Lesson 2

In calculus, the fundamental theorem is an essential tool that helps explain the relationship between integration and differentiation. Learn about evaluating definite integrals using the fundamental theorem, and work examples to gain understanding.

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