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

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

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integral of x^5sin(x)

integral

x^{5} 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))))) + C

Solution steps

\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)))))

Add a constant to the solution

= - 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))))) + C

\frac{1}{( x ^{2} ( x + 1 ) ^{2} )}

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

5^{- x}

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

\frac{1}{( x ^{2} + 1 ) ^{\frac{3}{2}}}

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