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

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

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integral of (sin(t)sqrt(1+cos(t)))

\int (sin (t) 1 + cos (t)) d t

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

Solution steps

\int sin (t) 1 + cos (t) d t

Apply u-substitution

: \int - u d u

= \int - u d u

Take the constant out:

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

= - \int u d u

Apply the Power Rule

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

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

Substitute back

u = 1 + cos (t)

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

Add a constant to the solution

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

y = 3 x - 2 x, (1, 1)

(2 e^{y} - x) \frac{d y}{d x} = 1

\frac{d}{d x} ((7 + x)^{5})

x \to - 0 lim (ln (x))

y = \frac{x ^{2} + 32 x}{8}, x = 4

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