Integral of x^2sin(x) dx
The solution
Detail solution
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=x2 and let dv(x)=sin(x).
Then du(x)=2x.
To find v(x):
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The integral of sine is negative cosine:
∫sin(x)dx=−cos(x)
Now evaluate the sub-integral.
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=−2x and let dv(x)=cos(x).
Then du(x)=−2.
To find v(x):
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The integral of cosine is sine:
∫cos(x)dx=sin(x)
Now evaluate the sub-integral.
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The integral of a constant times a function is the constant times the integral of the function:
∫(−2sin(x))dx=−2∫sin(x)dx
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The integral of sine is negative cosine:
∫sin(x)dx=−cos(x)
So, the result is: 2cos(x)
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Add the constant of integration:
−x2cos(x)+2xsin(x)+2cos(x)+constant
The answer is:
−x2cos(x)+2xsin(x)+2cos(x)+constant
The answer (Indefinite)
[src]
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| 2 2
| x *sin(x) dx = C + 2*cos(x) - x *cos(x) + 2*x*sin(x)
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∫x2sin(x)dx=C−x2cos(x)+2xsin(x)+2cos(x)
The graph
−2+cos(1)+2sin(1)
=
−2+cos(1)+2sin(1)
Use the examples entering the upper and lower limits of integration.