What is the integral from 0 to x of (cos(x)-sin(y)) ?

The integral from 0 to x of (cos(x)-sin(y)) is xcos(x)+cos(x)-1

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integral from 0 to x of (cos(x)-sin(y))

\int_{0}^{x} (cos (x) - sin (y)) d y

x cos (x) + cos (x) - 1

Solution steps

\int_{0}^{x} cos (x) - sin (y) d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{x} cos (x) d y - \int_{0}^{x} sin (y) d y

\int_{0}^{x} cos (x) d y = x cos (x)

\int_{0}^{x} sin (y) d y = - cos (x) + 1

= x cos (x) - (- cos (x) + 1)

Simplify

= x cos (x) + cos (x) - 1

What is the integral from 0 to x of (cos(x)-sin(y)) ?
The integral from 0 to x of (cos(x)-sin(y)) is xcos(x)+cos(x)-1