What is the integral of xcos((npix)/5) ?

The integral of xcos((npix)/5) is 1/(pi^2n^2)(5pinxsin((pinx)/5)+25cos((pinx)/5))+C

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integral of xcos((npix)/5)

\int x cos (\frac{nπ x}{5}) d x

\frac{1}{π ^{2} n ^{2}} (5 πn x sin (\frac{πn x}{5}) + 25 cos (\frac{πn x}{5})) + C

Solution steps

\int x cos (\frac{nπ x}{5}) d x

Apply u-substitution

: \int \frac{u cos ( \frac{u}{5} )}{π ^{2} n ^{2}} d u

= \int \frac{u cos ( \frac{u}{5} )}{π ^{2} n ^{2}} d u

Take the constant out:

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

= \frac{1}{π ^{2} n ^{2}} \cdot \int u cos (\frac{u}{5}) d u

Apply Integration By Parts

: 5 u sin (\frac{u}{5}) - \int 5 sin (\frac{u}{5}) d u

= \frac{1}{π ^{2} n ^{2}} (5 u sin (\frac{u}{5}) - \int 5 sin (\frac{u}{5}) d u)

\int 5 sin (\frac{u}{5}) d u = - 25 cos (\frac{u}{5})

= \frac{1}{π ^{2} n ^{2}} (5 u sin (\frac{u}{5}) - (- 25 cos (\frac{u}{5})))

Substitute back

u = nπ x

= \frac{1}{π ^{2} n ^{2}} (5 nπ x sin (\frac{nπ x}{5}) - (- 25 cos (\frac{nπ x}{5})))

Simplify

= \frac{1}{π ^{2} n ^{2}} (5 πn x sin (\frac{πn x}{5}) + 25 cos (\frac{πn x}{5}))

Add a constant to the solution

= \frac{1}{π ^{2} n ^{2}} (5 πn x sin (\frac{πn x}{5}) + 25 cos (\frac{πn x}{5})) + C

(1 + x)^{\frac{1}{6}}

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\frac{d}{d x} (\frac{x}{9 + x ^{2}})

f (x) = (6 x - x^{3}) (5 x + 2)

\int_{1}^{e^{5}} x^{3} ln (x) d x

What is the integral of xcos((npix)/5) ?
The integral of xcos((npix)/5) is 1/(pi^2n^2)(5pinxsin((pinx)/5)+25cos((pinx)/5))+C