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- Improper integral
- Double Integral
- Triple Integral
- Curvilinear integral
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- Differential equations Step by Step
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How to use it?
- Integral of d{x}:
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Integral of sinxcos2x
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Integral of (x(x^2-7))/(x^2+2)dx
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Integral of x^2/sqrt(1+x^4)
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Integral of (x^2)sin2x
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Identical expressions
- (t+ two)/(t^ two +2t+ five)
- (t plus 2) divide by (t squared plus 2t plus 5)
- (t plus two) divide by (t to the power of two plus 2t plus five)
- (t+2)/(t2+2t+5)
- t+2/t2+2t+5
- (t+2)/(t²+2t+5)
- (t+2)/(t to the power of 2+2t+5)
- t+2/t^2+2t+5
- (t+2) divide by (t^2+2t+5)
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Similar expressions
- (t-2)/(t^2+2t+5)
- (t+2)/(t^2+2t-5)
- (t+2)/(t^2-2t+5)
Integral of (t+2)/(t^2+2t+5) dx
The solution
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[src]
1 / | | t + 2 | ------------ dt | 2 | t + 2*t + 5 | / 0
$$\int\limits_{0}^{1} \frac{t + 2}{\left(t^{2} + 2 t\right) + 5}\, dt$$
Integral((t + 2)/(t^2 + 2*t + 5), (t, 0, 1))
Detail solution
We have the integral:
/ | | t + 2 | ------------ dt | 2 | t + 2*t + 5 | /
Rewrite the integrand
/ 2*t + 2 \
|------------|
| 2 |
t + 2 \t + 2*t + 5/ 1
------------ = -------------- + ------------------
2 2 / 2 \
t + 2*t + 5 |/ t 1\ |
4*||- - - -| + 1|
\\ 2 2/ /or
/ | | t + 2 | ------------ dt | 2 = | t + 2*t + 5 | /
/
|
/ | 1
| | -------------- dt
| 2*t + 2 | 2
| ------------ dt | / t 1\
| 2 | |- - - -| + 1
| t + 2*t + 5 | \ 2 2/
| |
/ /
------------------ + --------------------
2 4 In the integral
/
|
| 2*t + 2
| ------------ dt
| 2
| t + 2*t + 5
|
/
------------------
2 do replacement
2 u = t + 2*t
then
the integral =
/
|
| 1
| ----- du
| 5 + u
|
/ log(5 + u)
----------- = ----------
2 2 do backward replacement
/
|
| 2*t + 2
| ------------ dt
| 2
| t + 2*t + 5
| / 2 \
/ log\5 + t + 2*t/
------------------ = -----------------
2 2 In the integral
/
|
| 1
| -------------- dt
| 2
| / t 1\
| |- - - -| + 1
| \ 2 2/
|
/
--------------------
4 do replacement
1 t
v = - - - -
2 2then
the integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
4 4 do backward replacement
/
|
| 1
| -------------- dt
| 2
| / t 1\
| |- - - -| + 1
| \ 2 2/ /1 t\
| atan|- + -|
/ \2 2/
-------------------- = -----------
4 2 Solution is:
/1 t\
atan|- + -| / 2 \
\2 2/ log\5 + t + 2*t/
C + ----------- + -----------------
2 2
The answer (Indefinite)
[src]
/ /1 t\ | atan|- + -| / 2 \ | t + 2 \2 2/ log\5 + t + 2*t/ | ------------ dt = C + ----------- + ----------------- | 2 2 2 | t + 2*t + 5 | /
$$\int \frac{t + 2}{\left(t^{2} + 2 t\right) + 5}\, dt = C + \frac{\log{\left(t^{2} + 2 t + 5 \right)}}{2} + \frac{\operatorname{atan}{\left(\frac{t}{2} + \frac{1}{2} \right)}}{2}$$
The graph
The answer
[src]
log(8) atan(1/2) log(5) pi ------ - --------- - ------ + -- 2 2 2 8
$$- \frac{\log{\left(5 \right)}}{2} - \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2} + \frac{\pi}{8} + \frac{\log{\left(8 \right)}}{2}$$
=
=
log(8) atan(1/2) log(5) pi ------ - --------- - ------ + -- 2 2 2 8
$$- \frac{\log{\left(5 \right)}}{2} - \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2} + \frac{\pi}{8} + \frac{\log{\left(8 \right)}}{2}$$
log(8)/2 - atan(1/2)/2 - log(5)/2 + pi/8
The graph
Use the examples entering the upper and lower limits of integration.