"Use a table of integrals to evaluate the..." was answered by students like you

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Edmund Conti

Answered question

2021-12-03

Use a table of integrals to evaluate the following integrals.

\int_{0}^{\frac{π}{2}} \frac{d θ}{1 + \sin 2 θ}

Answer & Explanation

Alicia Washington

Beginner2021-12-04Added 23 answers

Step 1
Given: integral
$\int_{0}^{\frac{π}{2}} \frac{d θ}{1 + \sin 2 θ}$
Formula used:
$\frac{2}{(\sin θ + \cos θ)^{2}} = \sec^{2} (\frac{π}{4} - \frac{x}{2})$
Step 2
To solve the integral first simplify and use the formula,
$\int_{0}^{\frac{π}{2}} \frac{d θ}{1 + \sin 2 θ} = \int_{0}^{\frac{π}{2}} \frac{d θ}{\sin^{2} θ + \cos^{2} θ + 2 \sin θ \cos θ}$
$= \int_{0}^{\frac{π}{2}} \frac{d θ}{(\sin θ + \cos θ)^{2}}$
$= \frac{1}{2} \int_{0}^{\frac{π}{2}} (\frac{π}{4} - \frac{θ}{2}) d θ$
Step 3
Now, integrate the integral,
$\frac{1}{2} \int_{0}^{\frac{π}{2}} \sec^{2} (\frac{π}{4} - \frac{θ}{2}) d θ = \frac{1}{2} (- 2 \tan (\frac{π}{4} - \frac{θ}{2}))_{0}^{\frac{π}{2}}$
$= - (\tan (\frac{π}{4} - \frac{θ}{2}))_{0}^{\frac{π}{2}}$
$= - \tan (\frac{π}{4} - \frac{π / 2}{2}) + \tan (\frac{π}{4} - \frac{0}{2})$
$= - \tan 0 + \tan \frac{π}{4}$
=0+1
=1

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