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

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

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

\int 1 + sin (2 x) d x

sin (x) - cos (x) + C

Solution steps

\int 1 + sin (2 x) d x

Rewrite using trig identities

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

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

assuming

(cos (x) + sin (x)) \geq 0

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

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

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

= sin (x) - cos (x)

Add a constant to the solution

= sin (x) - cos (x) + C

f (x) = x^{2}, at x = 7

y^{^{''}} + 3 y^{^{'}} + 9 y = 0

h \to 6 lim (\frac{4 h}{h - 6})

\frac{d}{d x} (- tan (9 x))

x \to 0 lim ([(2 x + 1) (x - 3)])

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