What is the integral of (x^2+x+1)sin(x) ?

The integral of (x^2+x+1)sin(x) is -x^2cos(x)-xcos(x)+2xsin(x)+cos(x)+sin(x)+C

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integral of (x^2+x+1)sin(x)

\int (x^{2} + x + 1) sin (x) d x

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

Solution steps

\int (x^{2} + x + 1) sin (x) d x

Apply Integration By Parts

: - cos (x) (x^{2} + x + 1) - \int - cos (x) (2 x + 1) d x

= - cos (x) (x^{2} + x + 1) - \int - cos (x) (2 x + 1) d x

\int - cos (x) (2 x + 1) d x = - 2 x sin (x) - sin (x) - 2 cos (x)

= - cos (x) (x^{2} + x + 1) - (- 2 x sin (x) - sin (x) - 2 cos (x))

Simplify

- cos (x) (x^{2} + x + 1) - (- 2 x sin (x) - sin (x) - 2 cos (x)) : - x^{2} cos (x) - x cos (x) + 2 x sin (x) + cos (x) + sin (x)

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

Add a constant to the solution

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

x \to - 1 lim (1 + x)

\frac{d}{d x} (\frac{- 12 x}{( x ^{2} + 3 ) ^{2}})

2 x^{2} - 6 x + 3

\frac{\partial}{\partial x} (arccos (x y))

\int e^{- 3 x} tan (e^{- 3 x}) d x

What is the integral of (x^2+x+1)sin(x) ?
The integral of (x^2+x+1)sin(x) is -x^2cos(x)-xcos(x)+2xsin(x)+cos(x)+sin(x)+C