What is the integral of (3cos(x))/(sqrt(1+3sin(x))) ?

The integral of (3cos(x))/(sqrt(1+3sin(x))) is 2sqrt(1+3sin(x))+C

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integral of (3cos(x))/(sqrt(1+3sin(x)))

\int \frac{3 cos ( x )}{1 + 3 sin ( x )} d x

2 1 + 3 sin (x) + C

Solution steps

\int \frac{3 cos ( x )}{1 + 3 sin ( x )} d x

Take the constant out:

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

= 3 \cdot \int \frac{cos ( x )}{1 + 3 sin ( x )} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

= 3 \cdot \int \frac{1}{3 u} d u

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= 3 \cdot \frac{1}{3} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 + 3 sin (x)

= 3 \cdot \frac{1}{3} \cdot 2 (1 + 3 sin (x))^{\frac{1}{2}}

Simplify

3 \cdot \frac{1}{3} \cdot 2 (1 + 3 sin (x))^{\frac{1}{2}} : 2 1 + 3 sin (x)

= 2 1 + 3 sin (x)

Add a constant to the solution

= 2 1 + 3 sin (x) + C

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What is the integral of (3cos(x))/(sqrt(1+3sin(x))) ?
The integral of (3cos(x))/(sqrt(1+3sin(x))) is 2sqrt(1+3sin(x))+C