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How to use it?
- Integral of d{x}:
-
Integral of sin^3x/cosx
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Integral of 3*exp(-3*x)
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Integral of 2*x/(1+x)
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Integral of xe^-1
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Identical expressions
- (five -x)/(3x^ two + one)
- (5 minus x) divide by (3x squared plus 1)
- (five minus x) divide by (3x to the power of two plus one)
- (5-x)/(3x2+1)
- 5-x/3x2+1
- (5-x)/(3x²+1)
- (5-x)/(3x to the power of 2+1)
- 5-x/3x^2+1
- (5-x) divide by (3x^2+1)
- (5-x)/(3x^2+1)dx
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Similar expressions
- (5+x)/(3x^2+1)
- (5-x)/(3x^2-1)
Integral of (5-x)/(3x^2+1) dx
The solution
You have entered
[src]
1 / | | 5 - x | -------- dx | 2 | 3*x + 1 | / 0
$$\int\limits_{0}^{1} \frac{- x + 5}{3 x^{2} + 1}\, dx$$
Integral((5 - x)/(3*x^2 + 1), (x, 0, 1))
Detail solution
We have the integral:
/ | | 5 - x | 1*-------- dx | 2 | 3*x + 1 | /
Rewrite the integrand
/ 3*2*x + 0 \
|--------------| /5\
| 2 | |-|
5 - x \3*x + 0*x + 1/ \1/
-------- = - ---------------- + -------------------
2 6 2
3*x + 1 / ___ \
\-\/ 3 *x + 0/ + 1or
/ | | 5 - x | 1*-------- dx | 2 = | 3*x + 1 | /
/
|
| 3*2*x + 0
| -------------- dx
| 2
/ | 3*x + 0*x + 1
| |
| 1 /
5* | ------------------- dx - --------------------
| 2 6
| / ___ \
| \-\/ 3 *x + 0/ + 1
|
/ In the integral
/
|
| 3*2*x + 0
- | -------------- dx
| 2
| 3*x + 0*x + 1
|
/
----------------------
6 do replacement
2 u = 3*x
then
the integral =
/
|
| 1
- | ----- du
| 1 + u
|
/ -log(1 + u)
------------- = ------------
6 6 do backward replacement
/
|
| 3*2*x + 0
- | -------------- dx
| 2
| 3*x + 0*x + 1
| / 2\
/ -log\1 + 3*x /
---------------------- = ---------------
6 6 In the integral
/ | | 1 5* | ------------------- dx | 2 | / ___ \ | \-\/ 3 *x + 0/ + 1 | /
do replacement
___ v = -x*\/ 3
then
the integral =
/ | | 1 5* | ------ dv = 5*atan(v) | 2 | 1 + v | /
do backward replacement
/ | ___ / ___\ | 1 5*\/ 3 *atan\x*\/ 3 / 5* | ------------------- dx = --------------------- | 2 3 | / ___ \ | \-\/ 3 *x + 0/ + 1 | /
Solution is:
/1 2\
log|- + x | ___ / ___\
\3 / 5*\/ 3 *atan\x*\/ 3 /
C - ----------- + ---------------------
6 3
The answer (Indefinite)
[src]
/ | / 2\ ___ / ___\ | 5 - x log\1 + 3*x / 5*\/ 3 *atan\x*\/ 3 / | -------- dx = C - ------------- + --------------------- | 2 6 3 | 3*x + 1 | /
$${{5\,\arctan \left(\sqrt{3}\,x\right)}\over{\sqrt{3}}}-{{\log
\left(3\,x^2+1\right)}\over{6}}$$
The graph
The answer
[src]
___
log(3) log(4/3) 5*pi*\/ 3
- ------ - -------- + ----------
6 6 9
$${{5\,\pi}\over{3^{{{3}\over{2}}}}}-{{\log 4}\over{6}}$$
=
=
___
log(3) log(4/3) 5*pi*\/ 3
- ------ - -------- + ----------
6 6 9
$$- \frac{\log{\left(3 \right)}}{6} - \frac{\log{\left(\frac{4}{3} \right)}}{6} + \frac{5 \sqrt{3} \pi}{9}$$
The graph
Use the examples entering the upper and lower limits of integration.