Integral of arctg2x-(3/(x^4))arctg2x dx
The solution
Detail solution
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Integrate term-by-term:
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The integral of a constant times a function is the constant times the integral of the function:
∫(−x43atan(2x))dx=−∫x43atan(2x)dx
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The integral of a constant times a function is the constant times the integral of the function:
∫x43atan(2x)dx=3∫x4atan(2x)dx
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=atan(2x) and let dv(x)=x41.
Then du(x)=4x2+12.
To find v(x):
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The integral of xn is n+1xn+1 when n=−1:
∫x41dx=−3x31
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:
∫(−3x3(4x2+1)2)dx=−32∫x3(4x2+1)1dx
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There are multiple ways to do this integral.
Method #1
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Let u=x2.
Then let du=2xdx and substitute du:
∫8u3+2u21du
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Rewrite the integrand:
8u3+2u21=4u+18−u2+2u21
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Integrate term-by-term:
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The integral of a constant times a function is the constant times the integral of the function:
∫4u+18du=8∫4u+11du
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Let u=4u+1.
Then let du=4du and substitute 4du:
∫4u1du
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The integral of a constant times a function is the constant times the integral of the function:
∫u1du=4∫u1du
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The integral of u1 is log(u).
So, the result is: 4log(u)
Now substitute u back in:
4log(4u+1)
So, the result is: 2log(4u+1)
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The integral of a constant times a function is the constant times the integral of the function:
∫(−u2)du=−2∫u1du
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The integral of u1 is log(u).
So, the result is: −2log(u)
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The integral of a constant times a function is the constant times the integral of the function:
∫2u21du=2∫u21du
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The integral of un is n+1un+1 when n=−1:
∫u21du=−u1
So, the result is: −2u1
The result is: −2log(u)+2log(4u+1)−2u1
Now substitute u back in:
−2log(x2)+2log(4x2+1)−2x21
Method #2
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Rewrite the integrand:
x3(4x2+1)1=4x2+116x−x4+x31
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Integrate term-by-term:
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The integral of a constant times a function is the constant times the integral of the function:
∫4x2+116xdx=16∫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: 2log(4x2+1)
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The integral of a constant times a function is the constant times the integral of the function:
∫(−x4)dx=−4∫x1dx
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The integral of x1 is log(x).
So, the result is: −4log(x)
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The integral of xn is n+1xn+1 when n=−1:
∫x31dx=−2x21
The result is: −4log(x)+2log(4x2+1)−2x21
Method #3
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Rewrite the integrand:
x3(4x2+1)1=4x5+x31
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Rewrite the integrand:
4x5+x31=4x2+116x−x4+x31
-
Integrate term-by-term:
-
The integral of a constant times a function is the constant times the integral of the function:
∫4x2+116xdx=16∫4x2+1xdx
-
The integral of a constant times a function is the constant times the integral of the function:
∫4x2+1xdx=8∫4x2+18xdx
-
Let u=4x2+1.
Then let du=8xdx and substitute 8du:
∫8u1du
-
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: 2log(4x2+1)
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The integral of a constant times a function is the constant times the integral of the function:
∫(−x4)dx=−4∫x1dx
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The integral of x1 is log(x).
So, the result is: −4log(x)
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The integral of xn is n+1xn+1 when n=−1:
∫x31dx=−2x21
The result is: −4log(x)+2log(4x2+1)−2x21
Method #4
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Rewrite the integrand:
x3(4x2+1)1=4x5+x31
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Rewrite the integrand:
4x5+x31=4x2+116x−x4+x31
-
Integrate term-by-term:
-
The integral of a constant times a function is the constant times the integral of the function:
∫4x2+116xdx=16∫4x2+1xdx
-
The integral of a constant times a function is the constant times the integral of the function:
∫4x2+1xdx=8∫4x2+18xdx
-
Let u=4x2+1.
Then let du=8xdx and substitute 8du:
∫8u1du
-
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: 2log(4x2+1)
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−x4)dx=−4∫x1dx
-
The integral of x1 is log(x).
So, the result is: −4log(x)
-
The integral of xn is n+1xn+1 when n=−1:
∫x31dx=−2x21
The result is: −4log(x)+2log(4x2+1)−2x21
So, the result is: 34log(x2)−34log(4x2+1)+3x21
So, the result is: −4log(x2)+4log(4x2+1)−x21−x3atan(2x)
So, the result is: 4log(x2)−4log(4x2+1)+x21+x3atan(2x)
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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:
∫2atan(u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫atan(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
-
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)
The result is: xatan(2x)+4log(x2)−417log(4x2+1)+x21+x3atan(2x)
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Add the constant of integration:
xatan(2x)+4log(x2)−417log(4x2+1)+x21+x3atan(2x)+constant
The answer is:
xatan(2x)+4log(x2)−417log(4x2+1)+x21+x3atan(2x)+constant
The answer (Indefinite)
[src]
/
| / 2\
| / 3 \ 1 / 2\ 17*log\1 + 4*x / atan(2*x)
| |atan(2*x) - --*atan(2*x)| dx = C + -- + 4*log\x / - ---------------- + x*atan(2*x) + ---------
| | 4 | 2 4 3
| \ x / x x
|
/
∫(−x43atan(2x)+atan(2x))dx=C+xatan(2x)+4log(x2)−417log(4x2+1)+x21+x3atan(2x)
Use the examples entering the upper and lower limits of integration.