What is the integral of 2cos(npix) ?

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

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integral of 2cos(npix)

\int 2 cos (nπ x) d x

\frac{2}{πn} sin (πn x) + C

Solution steps

\int 2 cos (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 cos (nπ x) d x

Apply u-substitution

: \int \frac{cos ( u )}{πn} d u

= 2 \cdot \int \frac{cos ( 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 cos (u) d u

Use the common integral:

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

= 2 \cdot \frac{1}{πn} sin (u)

Substitute back

u = nπ x

= 2 \cdot \frac{1}{πn} sin (nπ x)

Simplify

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

= \frac{2}{πn} sin (πn x)

Add a constant to the solution

= \frac{2}{πn} sin (πn x) + C

\int sin (x) cos^{3} (x) d x

\int \frac{1}{5 - 4 x ^{2}} d x

\int_{0}^{2} 2 x - x^{3} d x

\int \frac{x + 5}{16 - ( x - 4 ) ^{2}} d x

f (x) = \frac{1}{5 x}

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