What is the integral of sin((npit)/2) ?

The integral of sin((npit)/2) is -2/(pin)cos((pin)/2 t)+C

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integral of sin((npit)/2)

\int sin (\frac{nπ t}{2}) d t

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

Solution steps

\int sin (\frac{nπ t}{2}) d t

\frac{nπ t}{2} = \frac{nπ}{2} t

= \int sin (\frac{nπ}{2} t) d t

Apply u-substitution

: \int \frac{2 sin ( u )}{πn} d u

= \int \frac{2 sin ( u )}{πn} d u

Take the constant out:

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

= \frac{2}{πn} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{2}{πn} (- cos (u))

Substitute back

u = \frac{nπ}{2} t

= \frac{2}{πn} (- cos (\frac{nπ}{2} t))

Simplify

= - \frac{2}{πn} cos (\frac{πn}{2} t)

Add a constant to the solution

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

\int (2 x^{2} + \frac{1}{( 2 x ) ^{2}}) d x

f (x) = - 2 x sin (x^{2})

\frac{\partial}{\partial c} (\frac{a ^{3} b ^{4}}{c ^{2} d ^{4}})

f (x) = arccos (x)

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

What is the integral of sin((npit)/2) ?
The integral of sin((npit)/2) is -2/(pin)cos((pin)/2 t)+C