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

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

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integral of 3sin(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

\frac{d}{d x} (3 + 2 x)

f (x) = \frac{x}{x - 2}, at x = 1

\int \frac{5 x ^{3} + 4 x}{x - 1} d x

\int \frac{x - 4}{x} d x

y^{^{'}} = \frac{7 x ^{6} e ^{\frac{y}{x}} + x ^{4} y ^{2}}{x ^{5} y}

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