What is the integral of (cos(x)+sin(x))^2 ?

The integral of (cos(x)+sin(x))^2 is x-1/2 cos(2x)+C

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integral of (cos(x)+sin(x))^2

\int (cos (x) + sin (x))^{2} d x

x - \frac{1}{2} cos (2 x) + C

Solution steps

\int (cos (x) + sin (x))^{2} d x

Rewrite using trig identities

= \int 1 + sin (2 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 1 d x + \int sin (2 x) d x

\int 1 d x = x

\int sin (2 x) d x = - \frac{1}{2} cos (2 x)

= x - \frac{1}{2} cos (2 x)

Add a constant to the solution

= x - \frac{1}{2} cos (2 x) + C

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What is the integral of (cos(x)+sin(x))^2 ?
The integral of (cos(x)+sin(x))^2 is x-1/2 cos(2x)+C