"Evaluate the integral absolute value of sin⁡(x+π) from..." was answered by our community

Secondary
Algebra
Algebra I
Piecewise-Defined Functions

Edmund Adams

Answered question

2021-12-07

Evaluate the integral absolute value of

\sin (x + π)

from 0 to

3 \frac{π}{2}

Answer & Explanation

menerkupvd

Beginner2021-12-08Added 12 answers

Step 1
To evaluate the integral

\int_{0}^{\frac{3 π}{2}} | \sin (x + π) | d x

Step 2

\int_{0}^{\frac{3 π}{2}} | \sin (x + π) | d x

= \int_{0}^{\frac{3 π}{2}} | \sin x \cos π + \cos x \sin π | d x

(using

\sin (a + b) = \sin a \cos b + \cos a \sin b

)

= \int_{0}^{\frac{3 π}{2}} | \sin x (- 1) + \cos x (0) | d x

= \int_{0}^{\frac{3 π}{2}} | - \sin x | d x

= \int_{0}^{\frac{3 π}{2}} \sin x d x

= {[- \cos x]}_{0}^{\frac{3 π}{2}}

= [- \cos (\frac{3 π}{2}) - (- c o s 0)]

=[-0]-[-1]
=1
Hence,

\int_{0}^{\frac{3 π}{2}} | \sin (x + π) | d x = 1

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