What is the integral of cos((npi)/L x) ?

The integral of cos((npi)/L x) is 1/(pin)Lsin((pinx)/L)+C

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integral of cos((npi)/L x)

\int cos (\frac{nπ}{L} x) d x

\frac{1}{πn} L sin (\frac{πn x}{L}) + C

Solution steps

\int cos (\frac{nπ}{L} x) d x

\frac{nπ}{L} = nπ \frac{1}{L}

= \int cos (nπ \frac{1}{L} x) d x

Multiply

nπ \frac{1}{L} x : \frac{πn x}{L}

= \int cos (\frac{πn x}{L}) d x

Apply u-substitution

: \int \frac{cos ( \frac{u}{L} )}{πn} d u

= \int \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 cos (\frac{u}{L}) d u

Apply u-substitution

: \int cos (v) L d v

= \frac{1}{πn} \cdot \int 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 cos (v) d v

Use the common integral:

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

= \frac{1}{πn} L sin (v)

Substitute back

= \frac{1}{πn} L sin (\frac{πn x}{L})

Add a constant to the solution

= \frac{1}{πn} L sin (\frac{πn x}{L}) + C

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What is the integral of cos((npi)/L x) ?
The integral of cos((npi)/L x) is 1/(pin)Lsin((pinx)/L)+C