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

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

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integral of x^2cos(nxpi)

\int x^{2} cos (n x π) 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 (n x π) 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 = n x π

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

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

\int \frac{1}{y 5 y ^{2} - 3} d y

\int (x - 0.1)^{2} (x - 0.75)^{2} d x

\int \frac{ln ( x )}{4} d x

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

\frac{\partial}{\partial x} (y^{0.75} x^{0.25})

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