Integral of xatan(x) dx
The solution
Detail solution
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=atan(x) and let dv(x)=x.
Then du(x)=x2+11.
To find v(x):
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The integral of xn is n+1xn+1 when n=−1:
∫xdx=2x2
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:
∫2(x2+1)x2dx=2∫x2+1x2dx
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Rewrite the integrand:
x2+1x2=1−x2+11
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Integrate term-by-term:
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The integral of a constant is the constant times the variable of integration:
∫1dx=x
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The integral of a constant times a function is the constant times the integral of the function:
∫(−x2+11)dx=−∫x2+11dx
PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), True), (ArccothRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False), (ArctanhRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False)], context=1/(x**2 + 1), symbol=x)
So, the result is: −atan(x)
The result is: x−atan(x)
So, the result is: 2x−2atan(x)
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Add the constant of integration:
2x2atan(x)−2x+2atan(x)+constant
The answer is:
2x2atan(x)−2x+2atan(x)+constant
The answer (Indefinite)
[src]
/ 2
| atan(x) x x *atan(x)
| x*atan(x) dx = C + ------- - - + ----------
| 2 2 2
/
∫xatan(x)dx=C+2x2atan(x)−2x+2atan(x)
The graph
−21+4π
=
−21+4π
Use the examples entering the upper and lower limits of integration.