What is the integral of-x^5sin(x) ?

The integral of-x^5sin(x) is x^5cos(x)-5x^4sin(x)-20x^3cos(x)+60x^2sin(x)+120xcos(x)-120sin(x)+C

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integral of-x^5sin(x)

\int - x^{5} sin (x) d x

x^{5} cos (x) - 5 x^{4} sin (x) - 20 x^{3} cos (x) + 60 x^{2} sin (x) + 120 x cos (x) - 120 sin (x) + C

Solution steps

\int - x^{5} sin (x) d x

Take the constant out:

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

= - \int x^{5} sin (x) d x

Apply Integration By Parts

: - x^{5} cos (x) - \int - 5 x^{4} cos (x) d x

= - (- x^{5} cos (x) - \int - 5 x^{4} cos (x) d x)

\int - 5 x^{4} cos (x) d x = - 5 (x^{4} sin (x) - 4 (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))))

= - (- x^{5} cos (x) - (- 5 (x^{4} sin (x) - 4 (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))))))

Simplify

- (- x^{5} cos (x) - (- 5 (x^{4} sin (x) - 4 (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))))))) : x^{5} cos (x) - 5 x^{4} sin (x) - 20 x^{3} cos (x) + 60 x^{2} sin (x) + 120 x cos (x) - 120 sin (x)

= x^{5} cos (x) - 5 x^{4} sin (x) - 20 x^{3} cos (x) + 60 x^{2} sin (x) + 120 x cos (x) - 120 sin (x)

Add a constant to the solution

= x^{5} cos (x) - 5 x^{4} sin (x) - 20 x^{3} cos (x) + 60 x^{2} sin (x) + 120 x cos (x) - 120 sin (x) + C

x \to 0 lim (500)

\frac{\partial}{\partial x} ((4 x^{3} y^{4} + 9)^{2})

\int_{- 3}^{5} ∣ x ∣ d x

\frac{\partial}{\partial x} (x^{3} y + y^{2})

\frac{1}{( s + 2 )}

What is the integral of-x^5sin(x) ?
The integral of-x^5sin(x) is x^5cos(x)-5x^4sin(x)-20x^3cos(x)+60x^2sin(x)+120xcos(x)-120sin(x)+C