What is the integral of x^2cos(npix) ?

The integral of x^2cos(npix) is 1/(pi^3n^3)(pi^2n^2x^2sin(pinx)-2(-pinxcos(pinx)+sin(pinx)))+C

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integral of x^2cos(npix)

\int x^{2} cos (nπ x) d x

\frac{1}{π ^{3} n ^{3}} (π^{2} n^{2} x^{2} sin (πn x) - 2 (- πn x cos (πn x) + sin (πn x))) + C

Solution steps

\int x^{2} cos (nπ x) d x

Apply u-substitution

: \int \frac{u ^{2} cos ( u )}{π ^{3} n ^{3}} d u

= \int \frac{u ^{2} cos ( u )}{π ^{3} n ^{3}} d u

Take the constant out:

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

= \frac{1}{π ^{3} n ^{3}} \cdot \int u^{2} cos (u) d u

Apply Integration By Parts

: u^{2} sin (u) - \int 2 u sin (u) d u

= \frac{1}{π ^{3} n ^{3}} (u^{2} sin (u) - \int 2 u sin (u) d u)

\int 2 u sin (u) d u = 2 (- u cos (u) + sin (u))

= \frac{1}{π ^{3} n ^{3}} (u^{2} sin (u) - 2 (- u cos (u) + sin (u)))

Substitute back

u = nπ x

= \frac{1}{π ^{3} n ^{3}} ((nπ x)^{2} sin (nπ x) - 2 (- nπ x cos (nπ x) + sin (nπ x)))

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= \frac{1}{π ^{3} n ^{3}} (π^{2} n^{2} x^{2} sin (πn x) - 2 (- πn x cos (πn x) + sin (πn x)))

Add a constant to the solution

= \frac{1}{π ^{3} n ^{3}} (π^{2} n^{2} x^{2} sin (πn x) - 2 (- πn x cos (πn x) + sin (πn x))) + C

cos (x^{2})

\int \frac{1}{sec ( x ^{2} )} d x

\int (e^{\frac{x}{a}} + 1)^{\frac{1}{3}} e^{\frac{x}{a}} d x

\int_{2}^{4} \frac{x ( 2 x - 3 )}{6} d x

x \to 1 - lim (\frac{8}{1 - x})

What is the integral of x^2cos(npix) ?
The integral of x^2cos(npix) is 1/(pi^3n^3)(pi^2n^2x^2sin(pinx)-2(-pinxcos(pinx)+sin(pinx)))+C