What is the integral of 2x^4sin(x) ?

The integral of 2x^4sin(x) is 2(-x^4cos(x)+4(x^3sin(x)-3(-x^2cos(x)+2(xsin(x)+cos(x)))))+C

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integral of 2x^4sin(x)

\int 2 x^{4} sin (x) d x

2 (- x^{4} cos (x) + 4 (x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))))) + C

Solution steps

\int 2 x^{4} sin (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x^{4} sin (x) d x

Apply Integration By Parts

: - x^{4} cos (x) - \int - 4 x^{3} cos (x) d x

= 2 (- x^{4} cos (x) - \int - 4 x^{3} cos (x) d x)

\int - 4 x^{3} cos (x) d x = - 4 (x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))))

= 2 (- x^{4} cos (x) - (- 4 (x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))))))

Simplify

= 2 (- x^{4} cos (x) + 4 (x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x)))))

Add a constant to the solution

= 2 (- x^{4} cos (x) + 4 (x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x))))) + C

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What is the integral of 2x^4sin(x) ?
The integral of 2x^4sin(x) is 2(-x^4cos(x)+4(x^3sin(x)-3(-x^2cos(x)+2(xsin(x)+cos(x)))))+C