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

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

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

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

2 sin (x) + 1 + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 u} d u

= \int \frac{1}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 2 sin (x) + 1

= \frac{1}{2} \cdot 2 (2 sin (x) + 1)^{\frac{1}{2}}

Simplify

\frac{1}{2} \cdot 2 (2 sin (x) + 1)^{\frac{1}{2}} : 2 sin (x) + 1

= 2 sin (x) + 1

Add a constant to the solution

= 2 sin (x) + 1 + C

\int x \cdot (4 - x) d x

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\int 4^{s i n (x)} cos (x) d x

x \to - 7 lim (\frac{x + 8}{x + 7})

\frac{d}{d x} (e^{x - 3})

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