What is the integral from 0 to L of cos((npix)/L) ?

The integral from 0 to L of cos((npix)/L) is 0

integral from 0 to L of cos((npix)/L)

\int_{0}^{L} cos (\frac{nπ x}{L}) d x

0

Solution steps

\int_{0}^{L} cos (\frac{nπ x}{L}) d x

Apply u-substitution

= \int_{0}^{n L π} \frac{cos ( \frac{u}{L} )}{πn} d u

Take the constant out:

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

= \frac{1}{πn} \cdot \int_{0}^{n L π} cos (\frac{u}{L}) d u

Apply u-substitution

= \frac{1}{πn} \cdot \int_{0}^{πn} cos (v) L d v

Take the constant out:

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

= \frac{1}{πn} L \cdot \int_{0}^{πn} cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= \frac{1}{πn} L [sin (v)]_{0}^{πn}

Compute the boundaries:

0

= \frac{1}{πn} L \cdot 0

Simplify

= 0

\int_{1}^{t} \frac{1}{x ^{2}} d x

\int_{2}^{10} x^{6} d x

\int_{x^{5}}^{x^{4}} (2 t - 1)^{3} d t

\int_{0}^{\frac{π}{2}} - sin (x) d x

\int_{1}^{5} \frac{2}{( x - 3 ) ^{4}} d x

What is the integral from 0 to L of cos((npix)/L) ?
The integral from 0 to L of cos((npix)/L) is 0