What is the integral of cos(npit) ?

The integral of cos(npit) is 1/(pin)sin(npit)+C

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integral of cos(npit)

\int cos (nπ t) d t

\frac{1}{πn} sin (nπ t) + C

Solution steps

\int cos (nπ t) d t

Apply u-substitution

: \int \frac{cos ( u )}{πn} d u

= \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

= \frac{1}{πn} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{πn} sin (u)

Substitute back

u = nπ t

= \frac{1}{πn} sin (nπ t)

Add a constant to the solution

= \frac{1}{πn} sin (nπ t) + C

11 x \cdot e^{x}

x \to 0 lim (tanh (x))

\frac{1}{2} (e^{x} - e^{- x})

\int \frac{x ^{n}}{n + 1} d x

What is the integral of cos(npit) ?
The integral of cos(npit) is 1/(pin)sin(npit)+C