What is the integral of t^3cos(t) ?

The integral of t^3cos(t) is t^3sin(t)-3(-t^2cos(t)+2(tsin(t)+cos(t)))+C

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integral of t^3cos(t)

\int t^{3} cos (t) d t

t^{3} sin (t) - 3 (- t^{2} cos (t) + 2 (t sin (t) + cos (t))) + C

Solution steps

\int t^{3} cos (t) d t

Apply Integration By Parts

: t^{3} sin (t) - \int 3 t^{2} sin (t) d t

= t^{3} sin (t) - \int 3 t^{2} sin (t) d t

\int 3 t^{2} sin (t) d t = 3 (- t^{2} cos (t) + 2 (t sin (t) + cos (t)))

= t^{3} sin (t) - 3 (- t^{2} cos (t) + 2 (t sin (t) + cos (t)))

Add a constant to the solution

= t^{3} sin (t) - 3 (- t^{2} cos (t) + 2 (t sin (t) + cos (t))) + C

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What is the integral of t^3cos(t) ?
The integral of t^3cos(t) is t^3sin(t)-3(-t^2cos(t)+2(tsin(t)+cos(t)))+C