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

The integral of sin(((npix))/2) is -2/(pin)cos((pin)/2 x)+C

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integral of sin(((npix))/2)

\int sin (\frac{( nπ x )}{2}) d x

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

Solution steps

\int sin (\frac{nπ x}{2}) d x

\frac{nπ x}{2} = \frac{nπ}{2} x

= \int sin (\frac{nπ}{2} x) d x

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} x

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

Simplify

= - \frac{2}{πn} cos (\frac{πn}{2} x)

Add a constant to the solution

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

\int \frac{1}{x sin ( y )} d x

\frac{d}{d x} (e^{s i n (c o s (x^{n}))})

\int (6 x^{2}) (2 x^{3} + 5)^{6} d x

\int \frac{2 x + 7}{x ^{2} + 3 x + 3} d x

\frac{d y}{d x} = \frac{( 2 x ( y - 1 ) )}{x ^{2} + 3}, y (1) = 9

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