What is the integral from x to pi-x of sin(x+y) ?

The integral from x to pi-x of sin(x+y) is 1+cos(2x)

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integral from x to pi-x of sin(x+y)

\int_{x}^{π - x} sin (x + y) d y

1 + cos (2 x)

Solution steps

\int_{x}^{π - x} sin (x + y) d y

Apply u-substitution

: \int_{2 x}^{π} sin (u) d u

= \int_{2 x}^{π} sin (u) d u

Use the common integral:

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

= [- cos (u)]_{2 x}^{π}

Compute the boundaries:

1 + cos (2 x)

= 1 + cos (2 x)

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What is the integral from x to pi-x of sin(x+y) ?
The integral from x to pi-x of sin(x+y) is 1+cos(2x)