Integral of arctan(2x) dx
The solution
Detail solution
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There are multiple ways to do this integral.
Method #1
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Let u=2x.
Then let du=2dx and substitute 2du:
∫4atan(u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫2atan(u)du=2∫atan(u)du
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Use integration by parts:
∫udv=uv−∫vdu
Let u(u)=atan(u) and let dv(u)=1.
Then du(u)=u2+11.
To find v(u):
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The integral of a constant is the constant times the variable of integration:
∫1du=u
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:
∫u2+1udu=2∫u2+12udu
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Let u=u2+1.
Then let du=2udu and substitute 2du:
∫2u1du
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The integral of u1 is log(u).
Now substitute u back in:
log(u2+1)
So, the result is: 2log(u2+1)
So, the result is: 2uatan(u)−4log(u2+1)
Now substitute u back in:
xatan(2x)−4log(4x2+1)
Method #2
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=atan(2x) and let dv(x)=1.
Then du(x)=4x2+12.
To find v(x):
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The integral of a constant is the constant times the variable of integration:
∫1dx=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:
∫4x2+12xdx=2∫4x2+1xdx
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The integral of a constant times a function is the constant times the integral of the function:
∫4x2+1xdx=8∫4x2+18xdx
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Let u=4x2+1.
Then let du=8xdx and substitute 8du:
∫8u1du
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The integral of u1 is log(u).
Now substitute u back in:
log(4x2+1)
So, the result is: 8log(4x2+1)
So, the result is: 4log(4x2+1)
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Add the constant of integration:
xatan(2x)−4log(4x2+1)+constant
The answer is:
xatan(2x)−4log(4x2+1)+constant
The answer (Indefinite)
[src]
/ / 2\
| log\1 + 4*x /
| atan(2*x) dx = C - ------------- + x*atan(2*x)
| 4
/
∫atan(2x)dx=C+xatan(2x)−4log(4x2+1)
The graph
log(5)
- ------ + atan(2)
4
−4log(5)+atan(2)
=
log(5)
- ------ + atan(2)
4
−4log(5)+atan(2)
Use the examples entering the upper and lower limits of integration.