Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x^2*e^(-x)
Integral of x^2*ln(x)
Integral of -4
Integral of 4x^2
Derivative of:
(x^2+3)/(x+1)
Graphing y =:
(x^2+3)/(x+1)
Identical expressions
(x^ two + three)/(x+ one)
(x squared plus 3) divide by (x plus 1)
(x to the power of two plus three) divide by (x plus one)
(x2+3)/(x+1)
x2+3/x+1
(x²+3)/(x+1)
(x to the power of 2+3)/(x+1)
x^2+3/x+1
(x^2+3) divide by (x+1)
(x^2+3)/(x+1)dx
Similar expressions
(x^2-3)/(x+1)
(x^2+3)/(x-1)

Solving integrals/ (x^2+3)/(x+1)

Integral of (x^2+3)/(x+1) dx

The solution

You have entered [src]

  1          
  /          
 |           
 |   2       
 |  x  + 3   
 |  ------ dx
 |  x + 1    
 |           
/            
0

$$\int\limits_{0}^{1} \frac{x^{2} + 3}{x + 1}\, dx$$

Integral((x^2 + 3)/(x + 1), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x^{2} + 3}{x + 1} = x - 1 + \frac{4}{x + 1}$
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:
    
    $\int \left(-1\right)\, dx = - x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{4}{x + 1}\, dx = 4 \int \frac{1}{x + 1}\, dx$
    1. Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 1 \right)}$
    So, the result is: $4 \log{\left(x + 1 \right)}$
  The result is: $\frac{x^{2}}{2} - x + 4 \log{\left(x + 1 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x^{2} + 3}{x + 1} = \frac{x^{2}}{x + 1} + \frac{3}{x + 1}$
2. Integrate term-by-term:
  1. Rewrite the integrand:
    
    $\frac{x^{2}}{x + 1} = x - 1 + \frac{1}{x + 1}$
  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:
      
      $\int \left(-1\right)\, dx = - x$
    1. Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 1 \right)}$
    The result is: $\frac{x^{2}}{2} - x + \log{\left(x + 1 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3}{x + 1}\, dx = 3 \int \frac{1}{x + 1}\, dx$
    1. Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 1 \right)}$
    So, the result is: $3 \log{\left(x + 1 \right)}$
  The result is: $\frac{x^{2}}{2} - x + \log{\left(x + 1 \right)} + 3 \log{\left(x + 1 \right)}$
Add the constant of integration:

$\frac{x^{2}}{2} - x + 4 \log{\left(x + 1 \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{2}}{2} - x + 4 \log{\left(x + 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                      
 |  2               2                   
 | x  + 3          x                    
 | ------ dx = C + -- - x + 4*log(1 + x)
 | x + 1           2                    
 |                                      
/

$$4\,\log \left(x+1\right)+{{x^2-2\,x}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-1/2 + 4*log(2)

$${{8\,\log 2-1}\over{2}}$$

-1/2 + 4*log(2)

$$- \frac{1}{2} + 4 \log{\left(2 \right)}$$

Simplify

Numerical answer [src]

2.27258872223978

2.27258872223978

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (x^2+3)/(x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2