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

The integral of t^2cos(nt) is 1/(n^3)(n^2t^2sin(nt)-2(-ntcos(nt)+sin(nt)))+C

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integral of t^2cos(nt)

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

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

Solution steps

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

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 = n t

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

Apply exponent rule:

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

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

Add a constant to the solution

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

\int \frac{- x}{x} d x

s \to 0 lim (s \cdot \frac{k}{s \cdot ( \frac{s}{5} + 1 ) \cdot ( \frac{s}{50} + 1 )})

\frac{\partial}{\partial x} (x^{3} y^{5} + 9 x^{5} y)

\int \frac{1}{9} (4 - x^{2}) d x

y = x^{3} - x^{2} - 6 x, x

What is the integral of t^2cos(nt) ?
The integral of t^2cos(nt) is 1/(n^3)(n^2t^2sin(nt)-2(-ntcos(nt)+sin(nt)))+C