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

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

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integral of sqrt(1+sin(θ))

\int 1 + sin (θ) d θ

2 (- cos (\frac{θ}{2}) + sin (\frac{θ}{2})) + C

Solution steps

\int 1 + sin (θ) d θ

Apply u-substitution

= \int \frac{2 ( u + 1 )}{( 1 + u ^{2} ) 1 + u ^{2}} d u

Take the constant out:

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

= 2 \cdot \int \frac{u + 1}{( 1 + u ^{2} ) 1 + u ^{2}} d u

Expand

\frac{u + 1}{( 1 + u ^{2} ) 1 + u ^{2}} : \frac{u}{( 1 + u ^{2} ) 1 + u ^{2}} + \frac{1}{( 1 + u ^{2} ) 1 + u ^{2}}

Apply the Sum Rule:

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

= 2 (\int \frac{u}{( 1 + u ^{2} ) 1 + u ^{2}} d u + \int \frac{1}{( 1 + u ^{2} ) 1 + u ^{2}} d u)

\int \frac{u}{( 1 + u ^{2} ) 1 + u ^{2}} d u = - \frac{1}{1 + u ^{2}}

\int \frac{1}{( 1 + u ^{2} ) 1 + u ^{2}} d u = \frac{u}{1 + u ^{2}}

= 2 (- \frac{1}{1 + u ^{2}} + \frac{u}{1 + u ^{2}})

Substitute back

u = tan (\frac{θ}{2})

= 2 - \frac{1}{1 + tan ^{2} ( \frac{θ}{2} )} + \frac{tan ( \frac{θ}{2} )}{1 + tan ^{2} ( \frac{θ}{2} )}

Simplify

2 - \frac{1}{1 + tan ^{2} ( \frac{θ}{2} )} + \frac{tan ( \frac{θ}{2} )}{1 + tan ^{2} ( \frac{θ}{2} )} : 2 (- cos (\frac{θ}{2}) + sin (\frac{θ}{2}))

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

Add a constant to the solution

= 2 (- cos (\frac{θ}{2}) + sin (\frac{θ}{2})) + C

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