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

The integral of sqrt(1-cos(t)) is -2sqrt(2)cos(t/2)+C

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integral of sqrt(1-cos(t))

\int 1 - cos (t) d t

- 22 cos (\frac{t}{2}) + C

Solution steps

\int 1 - cos (t) d t

Rewrite using trig identities

= \int 2 sin^{2} (\frac{t}{2}) d t

Take the constant out:

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

= 2 \cdot \int sin^{2} (\frac{t}{2}) d t

sin^{2} (\frac{t}{2}) = (sin (\frac{t}{2})),

assuming

sin (\frac{t}{2}) \geq 0

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

Apply u-substitution

: \int sin (u) \cdot 2 d u

= 2 \cdot \int sin (u) \cdot 2 d u

Take the constant out:

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

= 2 \cdot 2 \cdot \int sin (u) d u

Use the common integral:

\int sin (u) d u = - cos (u)

= 2 \cdot 2 (- cos (u))

Substitute back

u = \frac{t}{2}

= 2 \cdot 2 (- cos (\frac{t}{2}))

Simplify

= - 22 cos (\frac{t}{2})

Add a constant to the solution

= - 22 cos (\frac{t}{2}) + C

\int 13 + 12 x - x^{2} d x

\int \frac{x + 1}{- x ^{2} - 1} d x

y^{^{'}} = 2 y + x^{2} + 7

\frac{d ^{2}}{d x ^{2}} (e^{3 x})

\frac{1}{s ^{2} + 8} (- \frac{40}{33})

What is the integral of sqrt(1-cos(t)) ?
The integral of sqrt(1-cos(t)) is -2sqrt(2)cos(t/2)+C