What is the integral of 2sin(npix) ?

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

integral of 2sin(npix)

\int 2 sin (nπ x) d x

- \frac{2}{πn} cos (πn x) + C

Solution steps

\int 2 sin (nπ x) d x

Take the constant out:

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

= 2 \cdot \int sin (nπ x) d x

Apply u-substitution

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

Take the constant out:

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

= 2 \cdot \frac{1}{πn} \cdot \int sin (u) d u

Use the common integral:

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

= 2 \cdot \frac{1}{πn} (- cos (u))

Substitute back

u = nπ x

= 2 \cdot \frac{1}{πn} (- cos (nπ x))

Simplify

2 \cdot \frac{1}{πn} (- cos (nπ x)) : - \frac{2}{πn} cos (πn x)

= - \frac{2}{πn} cos (πn x)

Add a constant to the solution

= - \frac{2}{πn} cos (πn x) + C

\frac{d}{d x} ((x^{2} - x^{\frac{7}{3}})^{4})

10 y^{^{'}} + 5 y = 8 e^{6 x}, y (0) = 9

x \to 3 lim (\frac{∣ x ∣ - x}{3 - x})

y^{^{''}} + (y^{^{'}})^{2} + 1 = 0

\int_{- 40}^{0} \frac{1}{9 - x} d x

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