# Evaluate the integral. Integral from 0 to pi/4 of 7*sec(theta)*tan(theta) d(theta).

## Question:

Evaluate the integral.

{eq}\displaystyle \int_{0}^{\frac{\pi}{4}} 7 \sec(\theta) \tan(\theta) \, \mathrm{d}\theta {/eq}

## Answer and Explanation:

For this particular definite integral, define {eq}u {/eq} as follows:

{eq}u = \sec \theta {/eq}

Then:

{eq}\mathrm{d}u = \sec \theta \ \tan \theta \ \mathrm{d}\theta {/eq}

Plugging in {eq}\mathrm{d}u = \sec \theta \ \tan \theta \ \mathrm{d}\theta {/eq}:

{eq}\begin{align*} \displaystyle \int_{0}^{\frac{\pi}{4}} 7 \sec(\theta) \tan(\theta) \, \mathrm{d}\theta & =\displaystyle 7\int_{0}^{\frac{\pi}{4}} \sec(\theta) \tan(\theta) \, \mathrm{d}\theta\\ & =\displaystyle 7\int_{0}^{\frac{\pi}{4}} \mathrm{d}u \;\;\; \left[ \mathrm{ U-Substitution }\right]\\ & =7\left[ u \right]_{0}^{\frac{\pi}{4}} \\ & = 7\left[ \sec \theta \right]_{0}^{\frac{\pi}{4}} \;\;\; \left[ \mathrm{ Substitute \ back \ }u= \sec \theta\right]\\ & = 7 \left(\sec\left( \frac{\pi}{4}\right) - \sec 0 \right) \\ & = 7 \left(\sqrt{2}- 1 \right) \\ \implies \displaystyle \int_{0}^{\frac{\pi}{4}} 7 \sec(\theta) \tan(\theta) \, \mathrm{d}\theta& = 7 \sqrt{2}-7\\ \end{align*} {/eq}

#### Ask a question

Our experts can answer your tough homework and study questions.

Ask a question Ask a question#### Search Answers

#### Learn more about this topic:

#### Explore our homework questions and answer library

#### Our tutors are standing by

Ask a study question and one of our experts will send you an answer within hours.

To ask a site support question, click here

### Your question has been submitted!

When your answer is ready, it will appear on your Dashboard.

New! Get a text message when your answer is ready

Get Text AlertsNew! Get a text message when your answer is ready

Thanks! We'll text you when your answer is ready!