What is the integral of 5sin((npix)/4) ?

The integral of 5sin((npix)/4) is -(20}{pin}cos(\frac{pinx)/4)+C

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integral of 5sin((npix)/4)

\int 5 sin (\frac{nπ x}{4}) d x

- \frac{20}{πn} cos (\frac{πn x}{4}) + C

Solution steps

\int 5 sin (\frac{nπ x}{4}) d x

Take the constant out:

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

= 5 \cdot \int sin (\frac{nπ x}{4}) d x

Apply u-substitution

: \int \frac{4 sin ( u )}{πn} d u

= 5 \cdot \int \frac{4 sin ( u )}{πn} d u

Take the constant out:

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

= 5 \cdot \frac{4}{πn} \cdot \int sin (u) d u

Use the common integral:

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

= 5 \cdot \frac{4}{πn} (- cos (u))

Substitute back

u = \frac{n x π}{4}

= 5 \cdot \frac{4}{πn} (- cos (\frac{n x π}{4}))

Simplify

5 \cdot \frac{4}{πn} (- cos (\frac{n x π}{4})) : - \frac{20}{πn} cos (\frac{πn x}{4})

= - \frac{20}{πn} cos (\frac{πn x}{4})

Add a constant to the solution

= - \frac{20}{πn} cos (\frac{πn x}{4}) + C

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What is the integral of 5sin((npix)/4) ?
The integral of 5sin((npix)/4) is -(20}{pin}cos(\frac{pinx)/4)+C