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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of 2/x^4
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Integral of 1/5x
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Integral of x*dx/(x^2+1)
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Integral of x^3*exp(x^2)
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Identical expressions
- one /(one + two *x^ two *dx)
- 1 divide by (1 plus 2 multiply by x squared multiply by dx)
- one divide by (one plus two multiply by x to the power of two multiply by dx)
- 1/(1+2*x2*dx)
- 1/1+2*x2*dx
- 1/(1+2*x²*dx)
- 1/(1+2*x to the power of 2*dx)
- 1/(1+2x^2dx)
- 1/(1+2x2dx)
- 1/1+2x2dx
- 1/1+2x^2dx
- 1 divide by (1+2*x^2*dx)
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Similar expressions
- 1/(1-2*x^2*dx)
Integral of 1/(1+2*x^2*dx) dx
The solution
You have entered
[src]
1 / | | 1 | 1*---------- dx | 2 | 1 + 2*x *1 | / 0
$$\int\limits_{0}^{1} 1 \cdot \frac{1}{2 x^{2} \cdot 1 + 1}\, dx$$
Integral(1/(1 + 2*x^2*1), (x, 0, 1))
Detail solution
We have the integral:
/ | | 1 | 1*1*---------- dx | 2 | 1 + 2*x *1 | /
Rewrite the integrand
1 1
1*---------- = -----------------------
2 / 2 \
1 + 2*x *1 |/ ___ \ |
1*\\-\/ 2 *x + 0/ + 1/or
/ | | 1 | 1*1*---------- dx | 2 = | 1 + 2*x *1 | /
/ | | 1 | ------------------- dx | 2 | / ___ \ | \-\/ 2 *x + 0/ + 1 | /
In the integral
/ | | 1 | ------------------- dx | 2 | / ___ \ | \-\/ 2 *x + 0/ + 1 | /
do replacement
___ v = -x*\/ 2
then
the integral =
/ | | 1 | ------ dv = atan(v) | 2 | 1 + v | /
do backward replacement
/ | ___ / ___\ | 1 \/ 2 *atan\x*\/ 2 / | ------------------- dx = ------------------- | 2 2 | / ___ \ | \-\/ 2 *x + 0/ + 1 | /
Solution is:
___ / ___\
\/ 2 *atan\x*\/ 2 /
C + -------------------
2
The answer (Indefinite)
[src]
/ | ___ / ___\ | 1 \/ 2 *atan\x*\/ 2 / | 1*---------- dx = C + ------------------- | 2 2 | 1 + 2*x *1 | /
$${{\arctan \left(\sqrt{2}\,x\right)}\over{\sqrt{2}}}$$
The graph
The answer
[src]
___ / ___\
\/ 2 *atan\\/ 2 /
-----------------
2
$${{\arctan \sqrt{2}}\over{\sqrt{2}}}$$
=
=
___ / ___\
\/ 2 *atan\\/ 2 /
-----------------
2
$$\frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)}}{2}$$
The graph
Use the examples entering the upper and lower limits of integration.