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Similar expressions
1/x(x-1)^2

Integral of 1/x(x+1)^2 dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |    1        2   
 |  1*-*(x + 1)  dx
 |    x            
 |                 
/                  
0

$$\int\limits_{0}^{1} 1 \cdot \frac{1}{x} \left(x + 1\right)^{2}\, dx$$

Integral(1*(x + 1)^2/x, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $1 \cdot \frac{1}{x} \left(x + 1\right)^{2} = x + 2 + \frac{1}{x}$
2. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 2\, dx = 2 x$
  1. The integral of $\frac{1}{x}$ is $\log{\left(x \right)}$ .
  The result is: $\frac{x^{2}}{2} + 2 x + \log{\left(x \right)}$
Method #2
1. Rewrite the integrand:
  
  $1 \cdot \frac{1}{x} \left(x + 1\right)^{2} = \frac{x^{2} + 2 x + 1}{x}$
2. Rewrite the integrand:
  
  $\frac{x^{2} + 2 x + 1}{x} = x + 2 + \frac{1}{x}$
3. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 2\, dx = 2 x$
  1. The integral of $\frac{1}{x}$ is $\log{\left(x \right)}$ .
  The result is: $\frac{x^{2}}{2} + 2 x + \log{\left(x \right)}$
Add the constant of integration:

$\frac{x^{2}}{2} + 2 x + \log{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{2}}{2} + 2 x + \log{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
 |                        2               
 |   1        2          x                
 | 1*-*(x + 1)  dx = C + -- + 2*x + log(x)
 |   x                   2                
 |                                        
/

$$\log x+{{x^2+4\,x}\over{2}}$$

Simplify

The answer [src]

oo

$${\it \%a}$$

oo

$$\infty$$

Simplify

Numerical answer [src]

46.5904461339929

46.5904461339929

Use the examples entering the upper and lower limits of integration.

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Identical expressions

Similar expressions

Integral of 1/x(x+1)^2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2