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

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

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

\int 3 (sin (x)) 1 + cos (x) d x

- 2 (1 + cos (x))^{\frac{3}{2}} + C

Solution steps

\int 3 sin (x) 1 + cos (x) d x

Take the constant out:

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

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

Apply u-substitution

: \int - u d u

= 3 \cdot \int - u d u

Take the constant out:

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

= 3 (- \int u d u)

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= 3 (- \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = 1 + cos (x)

= 3 (- \frac{2}{3} (1 + cos (x))^{\frac{3}{2}})

Simplify

3 (- \frac{2}{3} (1 + cos (x))^{\frac{3}{2}}) : - 2 (1 + cos (x))^{\frac{3}{2}}

= - 2 (1 + cos (x))^{\frac{3}{2}}

Add a constant to the solution

= - 2 (1 + cos (x))^{\frac{3}{2}} + C

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