What is the integral of (x-sin(x)+cos(x)) ?

The integral of (x-sin(x)+cos(x)) is (x^2)/2+cos(x)+sin(x)+C

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integral of (x-sin(x)+cos(x))

\int (x - sin (x) + cos (x)) d x

\frac{x ^{2}}{2} + cos (x) + sin (x) + C

Solution steps

\int x - sin (x) + cos (x) d x

Apply the Sum Rule:

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

= \int x d x - \int sin (x) d x + \int cos (x) d x

\int x d x = \frac{x ^{2}}{2}

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

\int cos (x) d x = sin (x)

= \frac{x ^{2}}{2} - (- cos (x)) + sin (x)

Simplify

= \frac{x ^{2}}{2} + cos (x) + sin (x)

Add a constant to the solution

= \frac{x ^{2}}{2} + cos (x) + sin (x) + C

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What is the integral of (x-sin(x)+cos(x)) ?
The integral of (x-sin(x)+cos(x)) is (x^2)/2+cos(x)+sin(x)+C