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

\frac{d ^{2} y}{d x ^{2}} + 5 \frac{d y}{d x} + 4 y = 0

\frac{d y}{d x} = y^{2} (x + 1)

\int \frac{1}{9 e ^{y} + e ^{- y}} d y

\frac{d}{d x} (\frac{5 x ^{2}}{sin ( x )})

n = 1 \sum \infty \frac{2 ^{n}}{n ^{3}}

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