What is the integral from 0 to pi/3 of sin(x+y) ?

The integral from 0 to pi/3 of sin(x+y) is -cos(x+pi/3)+cos(x)

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integral from 0 to pi/3 of sin(x+y)

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

- cos (x + \frac{π}{3}) + cos (x)

Solution steps

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

Apply u-substitution

: \int_{x}^{x + \frac{π}{3}} sin (u) d u

= \int_{x}^{x + \frac{π}{3}} sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= [- cos (u)]_{x}^{x + \frac{π}{3}}

Compute the boundaries:

- cos (x + \frac{π}{3}) + cos (x)

= - cos (x + \frac{π}{3}) + cos (x)

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What is the integral from 0 to pi/3 of sin(x+y) ?
The integral from 0 to pi/3 of sin(x+y) is -cos(x+pi/3)+cos(x)