What is the Integral of x^2 sin pi x?

In summary, the integral of x^2 sin pi x is equal to (-1/2) x^2 cos pi x + (1/4) x sin pi x + (1/8) cos pi x + C, and can be solved using integration by parts or trigonometric substitution. The constant of integration is necessary to account for all possible solutions, and the domain of the integral is all real numbers. There is a graphical representation of the integral as the area under the curve y = x^2 sin pi x, bounded by the x-axis and the vertical lines x = 0 and x = 1.
  • #1
MillerGenuine
64
0
My calc is a bit rusty & i can not solve this problem for the life of me.
its the integral of x^2sin pi x.
I know you must integrate by parts..& i am have tried using both x^2 and sin pi x as my u.
Any help?
time is of the essence.


 
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  • #2
You have to integrate by parts twice. The trig part doesn't go away, but the polynomial part will go down in degree every time you differentiate it until you're just left with the integral of sin x.
 

What is the integral of x^2 sin pi x?

The integral of x^2 sin pi x is equal to (-1/2) x^2 cos pi x + (1/4) x sin pi x + (1/8) cos pi x + C, where C is the constant of integration.

How do you solve for the integral of x^2 sin pi x?

To solve for the integral of x^2 sin pi x, you can use integration by parts with u = x^2 and dv = sin pi x, or you can use the trigonometric substitution method with u = pi x and du = pi dx.

What is the significance of the constant of integration in the integral of x^2 sin pi x?

The constant of integration represents the unknown value that is added to the result of the integral to make it a general solution. It is necessary because the derivative of a constant is always equal to zero, so adding a constant ensures that all possible solutions are accounted for.

What is the domain of the integral of x^2 sin pi x?

The domain of the integral of x^2 sin pi x is all real numbers.

Is there a graphical representation of the integral of x^2 sin pi x?

Yes, the integral of x^2 sin pi x can be represented graphically as the area under the curve y = x^2 sin pi x, bounded by the x-axis and the vertical lines x = 0 and x = 1. This area can also be visualized as the shaded region in the first quadrant of the coordinate plane.

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