What is the integral of t^2cos(npit) ?

The integral of t^2cos(npit) is 1/(pi^3n^3)(pi^2n^2t^2sin(pint)-2(-pintcos(pint)+sin(pint)))+C

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integral of t^2cos(npit)

\int t^{2} cos (nπ t) d t

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

Solution steps

\int t^{2} cos (nπ t) d t

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 t

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

Apply exponent rule:

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

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

Add a constant to the solution

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

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What is the integral of t^2cos(npit) ?
The integral of t^2cos(npit) is 1/(pi^3n^3)(pi^2n^2t^2sin(pint)-2(-pintcos(pint)+sin(pint)))+C