What is the integral of sin((n*pi)/3 x) ?

The integral of sin((n*pi)/3 x) is -3/(pin)cos((pinx)/3)+C

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integral of sin((n*pi)/3 x)

\int sin (\frac{n \cdot π}{3} x) d x

- \frac{3}{πn} cos (\frac{πn x}{3}) + C

Solution steps

\int sin (\frac{nπ}{3} x) d x

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \int sin (\frac{πn x}{3}) d x

Apply u-substitution

: \int \frac{3 sin ( u )}{πn} d u

= \int \frac{3 sin ( u )}{πn} d u

Take the constant out:

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

= \frac{3}{πn} \cdot \int sin (u) d u

Use the common integral:

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

= \frac{3}{πn} (- cos (u))

Substitute back

u = \frac{n x π}{3}

= \frac{3}{πn} (- cos (\frac{n x π}{3}))

Simplify

= - \frac{3}{πn} cos (\frac{πn x}{3})

Add a constant to the solution

= - \frac{3}{πn} cos (\frac{πn x}{3}) + C

\int \frac{y}{1 + x y} d x

y = 1 - x^{2}, x = 2

\frac{x}{4 x + 8}

(x^{2} - 9) \frac{d y}{d x} + 6 y = (x + 3)^{2}

What is the integral of sin((n*pi)/3 x) ?
The integral of sin((n*pi)/3 x) is -3/(pin)cos((pinx)/3)+C