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

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

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integral of (x^2+x)cos(x)

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

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

Solution steps

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

Expand

(x^{2} + x) cos (x) : x^{2} cos (x) + x cos (x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int x cos (x) d x = x sin (x) + cos (x)

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

Add a constant to the solution

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

\int 1 + 5 x^{4} (x^{3}) d x

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\int \frac{e ^{x} e ^{x} - 1}{e ^{x} + 8} d x

\int e^{- x} + \frac{ln ( x )}{x} d x

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