What is the integral from 0 to 1 of x^2cos(nx) ?

The integral from 0 to 1 of x^2cos(nx) is (n^2sin(n)-2(sin(n)-ncos(n)))/(n^3)

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integral from 0 to 1 of x^2cos(nx)

\int_{0}^{1} x^{2} cos (n x) d x

\frac{n ^{2} sin ( n ) - 2 ( sin ( n ) - n cos ( n ) )}{n ^{3}}

Solution steps

\int_{0}^{1} x^{2} cos (n x) d x

Apply u-substitution

: \int_{0}^{n} \frac{u ^{2} cos ( u )}{n ^{3}} d u

= \int_{0}^{n} \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_{0}^{n} u^{2} cos (u) d u

Apply Integration By Parts

: [u^{2} sin (u) - \int 2 u sin (u) d u]_{0}^{n}

= \frac{1}{n ^{3}} [u^{2} sin (u) - \int 2 u sin (u) d u]_{0}^{n}

\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))]_{0}^{n}

Compute the boundaries:

n^{2} sin (n) - 2 (sin (n) - n cos (n))

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

Simplify

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

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What is the integral from 0 to 1 of x^2cos(nx) ?
The integral from 0 to 1 of x^2cos(nx) is (n^2sin(n)-2(sin(n)-ncos(n)))/(n^3)