What is the integral of 2(cos(x))^{-1/2}sin(x) ?

The integral of 2(cos(x))^{-1/2}sin(x) is -4cos^{1/2}(x)+C

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integral of 2(cos(x))^{-1/2}sin(x)

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

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

Solution steps

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

Take the constant out:

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

= 2 \cdot \int (cos (x))^{- \frac{1}{2}} sin (x) d x

Factor

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

= 2 \cdot \int \frac{sin ( x )}{cos ^{\frac{1}{2}} ( x )} d x

Apply u-substitution

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

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

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

Add a constant to the solution

= - 4 cos^{\frac{1}{2}} (x) + C

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What is the integral of 2(cos(x))^{-1/2}sin(x) ?
The integral of 2(cos(x))^{-1/2}sin(x) is -4cos^{1/2}(x)+C