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

The integral of x^2cos(xn) is 1/(n^3)(n^2x^2sin(nx)-2(-nxcos(nx)+sin(nx)))+C

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integral of x^2cos(xn)

\int x^{2} cos (x n) d x

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

Solution steps

\int x^{2} cos (x n) d x

Apply u-substitution

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

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

Take the constant out:

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

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

Substitute back

u = x n

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

Apply exponent rule:

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

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

Add a constant to the solution

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

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What is the integral of x^2cos(xn) ?
The integral of x^2cos(xn) is 1/(n^3)(n^2x^2sin(nx)-2(-nxcos(nx)+sin(nx)))+C