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

The integral of (2sin(x))/(sqrt(cos(x))) is -4sqrt(cos(x))+C

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

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

- 4 cos (x) + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = cos (x)

= 2 (- 2 cos^{\frac{1}{2}} (x))

Simplify

2 (- 2 cos^{\frac{1}{2}} (x)) : - 4 cos (x)

= - 4 cos (x)

Add a constant to the solution

= - 4 cos (x) + C

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What is the integral of (2sin(x))/(sqrt(cos(x))) ?
The integral of (2sin(x))/(sqrt(cos(x))) is -4sqrt(cos(x))+C