Evaluate the integral: {eq}\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \, (2 \sec \theta \tan \theta) \, \mathrm{d}\theta {/eq}.
Question:
Evaluate the integral: {eq}\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \, (2 \sec \theta \tan \theta) \, \mathrm{d}\theta {/eq}.
Definite integral with the secant and tangent functions.
This integral can be evaluated using the fundamental theorem of calculus.
{eq}\displaystyle\int_{a}^{b} f(\theta ) d \theta =F(b)-F(a), \ \text{if} \ F'(\theta)=f(\theta) \ \text{on} \ a\leq \theta\leq b {/eq}
We can use the FTOC on this problem because the anti-derivative of {eq}\, \sec(\theta)\tan(\theta) \, {/eq} is {eq}\, \sec(\theta) {/eq}.
{eq}\displaystyle\int \sec(\theta )\tan(\theta )\,\mathrm{d}\theta =\sec(\theta )+c {/eq}
Answer and Explanation: 1
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View this answer{eq}\begin{align*} \int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \, (2 \sec \theta \tan \theta) \, \mathrm{d}\theta&=\displaystyle\left [ 2\sec(\theta )...
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Chapter 24 / Lesson 2In this lesson you will learn how the fundamental theorem of calculus allows us to represent definite integrals as antiderivatives. You'll then cement their relationship in your mind by working through an example problem.