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