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