What is the integral of x^3sin(x) ?

The integral of x^3sin(x) is -x^3cos(x)+3(x^2sin(x)-2(-xcos(x)+sin(x)))+C

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integral of x^3sin(x)

\int x^{3} sin (x) d x

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

Solution steps

\int x^{3} sin (x) d x

Apply Integration By Parts

: - x^{3} cos (x) - \int - 3 x^{2} cos (x) d x

= - x^{3} cos (x) - \int - 3 x^{2} cos (x) d x

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

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

Simplify

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

Add a constant to the solution

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

\frac{\partial}{\partial t} (t^{4} e^{- x})

\frac{2}{1 - 2 x}

x \to 4 lim (\frac{x - 5}{x + 1})

\frac{\partial}{\partial x} ((2 x - 1)^{3})

9 y^{^{''}} + 6 y = 0

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