What is the integral of 4cos((npix)/4) ?

The integral of 4cos((npix)/4) is (16}{pin}sin(\frac{pinx)/4)+C

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integral of 4cos((npix)/4)

\int 4 cos (\frac{nπ x}{4}) d x

\frac{16}{πn} sin (\frac{πn x}{4}) + C

Solution steps

\int 4 cos (\frac{nπ x}{4}) d x

Take the constant out:

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

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

Apply Integration By Parts

= 4 (x cos (\frac{πn x}{4}) - \int - \frac{πn}{4} x sin (\frac{πn x}{4}) d x)

\int - \frac{πn}{4} x sin (\frac{πn x}{4}) d x = x cos (\frac{πn x}{4}) - \frac{4}{πn} sin (\frac{πn x}{4})

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

Simplify

4 (x cos (\frac{πn x}{4}) - (x cos (\frac{πn x}{4}) - \frac{4}{πn} sin (\frac{πn x}{4}))) : \frac{16}{πn} sin (\frac{πn x}{4})

= \frac{16}{πn} sin (\frac{πn x}{4})

Add a constant to the solution

= \frac{16}{πn} sin (\frac{πn x}{4}) + C

What is the integral of 4cos((npix)/4) ?
The integral of 4cos((npix)/4) is (16}{pin}sin(\frac{pinx)/4)+C