What is the integral of sqrt(1+cos(2x)) ?

The integral of sqrt(1+cos(2x)) is sqrt(2)sin(x)+C

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integral of sqrt(1+cos(2x))

\int 1 + cos (2 x) d x

2 sin (x) + C

Solution steps

\int 1 + cos (2 x) d x

Rewrite using trig identities

= \int 2 cos^{2} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int cos^{2} (x) d x

cos^{2} (x) = (cos (x)),

assuming

cos (x) \geq 0

= 2 \cdot \int cos (x) d x

Use the common integral:

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

= 2 sin (x)

Add a constant to the solution

= 2 sin (x) + C

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