Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of sin(x)*lnx
Integral of cos(3x)
Integral of x*exp(-5*x)
Integral of (x^4)e^(-(x^5)/4)
Identical expressions
(x²+ one)/((x²+ one)(x+ three))dx
(x² plus 1) divide by ((x² plus 1)(x plus 3))dx
(x² plus one) divide by ((x² plus one)(x plus three))dx
x²+1/x²+1x+3dx
(x²+1) divide by ((x²+1)(x+3))dx
Similar expressions
(x²+1)/((x²-1)(x+3))dx
(x²+1)/((x²+1)(x-3))dx
(x²-1)/((x²+1)(x+3))dx

Solving integrals/ (x²+1)/((x²+1)(x+3))dx

Integral of (x²+1)/((x²+1)(x+3))dx dx

The solution

You have entered [src]

  1                    
  /                    
 |                     
 |        2            
 |       x  + 1        
 |  ---------------- dx
 |  / 2    \           
 |  \x  + 1/*(x + 3)   
 |                     
/                      
0

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

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

Detail solution

Rewrite the integrand:

$\frac{x^{2} + 1}{\left(x + 3\right) \left(x^{2} + 1\right)} = \frac{x^{2}}{x^{3} + 3 x^{2} + x + 3} + \frac{1}{x^{3} + 3 x^{2} + x + 3}$
Integrate term-by-term:
1. Rewrite the integrand:
  
  $\frac{x^{2}}{x^{3} + 3 x^{2} + x + 3} = \frac{x - 3}{10 \left(x^{2} + 1\right)} + \frac{9}{10 \left(x + 3\right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{x - 3}{10 \left(x^{2} + 1\right)}\, dx = \frac{\int \frac{x - 3}{x^{2} + 1}\, dx}{10}$
    1. Rewrite the integrand:
      
      $\frac{x - 3}{x^{2} + 1} = \frac{x}{x^{2} + 1} - \frac{3}{x^{2} + 1}$
    2. Integrate term-by-term:
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}$
        
        Let $u = x^{2} + 1$ .
        
        Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
        
        $\int \frac{1}{2 u}\, du$
        
        The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
        
        Now substitute $u$ back in:
        
        $\log{\left(x^{2} + 1 \right)}$
        
        So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2}$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \left(- \frac{3}{x^{2} + 1}\right)\, dx = - 3 \int \frac{1}{x^{2} + 1}\, dx$
        
        PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), True), (ArccothRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False), (ArctanhRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False)], context=1/(x**2 + 1), symbol=x)
        
        So, the result is: $- 3 \operatorname{atan}{\left(x \right)}$
      The result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2} - 3 \operatorname{atan}{\left(x \right)}$
    So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{20} - \frac{3 \operatorname{atan}{\left(x \right)}}{10}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{9}{10 \left(x + 3\right)}\, dx = \frac{9 \int \frac{1}{x + 3}\, dx}{10}$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\log{\left(x + 3 \right)}$
    So, the result is: $\frac{9 \log{\left(x + 3 \right)}}{10}$
  The result is: $\frac{9 \log{\left(x + 3 \right)}}{10} + \frac{\log{\left(x^{2} + 1 \right)}}{20} - \frac{3 \operatorname{atan}{\left(x \right)}}{10}$
1. Rewrite the integrand:
  
  $\frac{1}{x^{3} + 3 x^{2} + x + 3} = - \frac{x - 3}{10 \left(x^{2} + 1\right)} + \frac{1}{10 \left(x + 3\right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{x - 3}{10 \left(x^{2} + 1\right)}\right)\, dx = - \frac{\int \frac{x - 3}{x^{2} + 1}\, dx}{10}$
    1. Rewrite the integrand:
      
      $\frac{x - 3}{x^{2} + 1} = \frac{x}{x^{2} + 1} - \frac{3}{x^{2} + 1}$
    2. Integrate term-by-term:
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{x}{x^{2} + 1}\, dx = \frac{\int \frac{2 x}{x^{2} + 1}\, dx}{2}$
        
        Let $u = x^{2} + 1$ .
        
        Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
        
        $\int \frac{1}{2 u}\, du$
        
        The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
        
        Now substitute $u$ back in:
        
        $\log{\left(x^{2} + 1 \right)}$
        
        So, the result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2}$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \left(- \frac{3}{x^{2} + 1}\right)\, dx = - 3 \int \frac{1}{x^{2} + 1}\, dx$
        
        PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), True), (ArccothRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False), (ArctanhRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False)], context=1/(x**2 + 1), symbol=x)
        
        So, the result is: $- 3 \operatorname{atan}{\left(x \right)}$
      The result is: $\frac{\log{\left(x^{2} + 1 \right)}}{2} - 3 \operatorname{atan}{\left(x \right)}$
    So, the result is: $- \frac{\log{\left(x^{2} + 1 \right)}}{20} + \frac{3 \operatorname{atan}{\left(x \right)}}{10}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{10 \left(x + 3\right)}\, dx = \frac{\int \frac{1}{x + 3}\, dx}{10}$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\log{\left(x + 3 \right)}$
    So, the result is: $\frac{\log{\left(x + 3 \right)}}{10}$
  The result is: $\frac{\log{\left(x + 3 \right)}}{10} - \frac{\log{\left(x^{2} + 1 \right)}}{20} + \frac{3 \operatorname{atan}{\left(x \right)}}{10}$
The result is: $\log{\left(x + 3 \right)}$
Add the constant of integration:

$\log{\left(x + 3 \right)}+ \mathrm{constant}$

The answer is:

$\log{\left(x + 3 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$\int \frac{x^{2} + 1}{\left(x + 3\right) \left(x^{2} + 1\right)}\, dx = C + \log{\left(x + 3 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-log(3) + log(4)

$$- \log{\left(3 \right)} + \log{\left(4 \right)}$$

-log(3) + log(4)

$$- \log{\left(3 \right)} + \log{\left(4 \right)}$$

-log(3) + log(4)

Numerical answer [src]

0.287682072451781

0.287682072451781

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 (x²+1)/((x²+1)(x+3))dx dx

Limits of integration:

The graph:

Piecewise:

The solution