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Integral of d{x}:
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Integral of x^2dx
Integral of sin²(x)
Integral of sec^2
Identical expressions
one /arctg two x(one +4x^2)
1 divide by arctg2x(1 plus 4x squared )
one divide by arctg two x(one plus 4x squared )
1/arctg2x(1+4x2)
1/arctg2x1+4x2
1/arctg2x(1+4x²)
1/arctg2x(1+4x to the power of 2)
1/arctg2x1+4x^2
1 divide by arctg2x(1+4x^2)
1/arctg2x(1+4x^2)dx
Similar expressions
1/arctg2x(1-4x^2)

Integral of 1/arctg2x(1+4x^2) dx

The solution

You have entered [src]

  1             
  /             
 |              
 |          2   
 |   1 + 4*x    
 |  --------- dx
 |  atan(2*x)   
 |              
/               
0

$$\int\limits_{0}^{1} \frac{4 x^{2} + 1}{\operatorname{atan}{\left(2 x \right)}}\, dx$$

Integral((1 + 4*x^2)/atan(2*x), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{4 x^{2} + 1}{\operatorname{atan}{\left(2 x \right)}} = \frac{4 x^{2}}{\operatorname{atan}{\left(2 x \right)}} + \frac{1}{\operatorname{atan}{\left(2 x \right)}}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{4 x^{2}}{\operatorname{atan}{\left(2 x \right)}}\, dx = 4 \int \frac{x^{2}}{\operatorname{atan}{\left(2 x \right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \frac{x^{2}}{\operatorname{atan}{\left(2 x \right)}}\, dx$
  So, the result is: $4 \int \frac{x^{2}}{\operatorname{atan}{\left(2 x \right)}}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\int \frac{1}{\operatorname{atan}{\left(2 x \right)}}\, dx$
The result is: $4 \int \frac{x^{2}}{\operatorname{atan}{\left(2 x \right)}}\, dx + \int \frac{1}{\operatorname{atan}{\left(2 x \right)}}\, dx$
Add the constant of integration:

$4 \int \frac{x^{2}}{\operatorname{atan}{\left(2 x \right)}}\, dx + \int \frac{1}{\operatorname{atan}{\left(2 x \right)}}\, dx+ \mathrm{constant}$

The answer is:

$4 \int \frac{x^{2}}{\operatorname{atan}{\left(2 x \right)}}\, dx + \int \frac{1}{\operatorname{atan}{\left(2 x \right)}}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$\int \frac{4 x^{2} + 1}{\operatorname{atan}{\left(2 x \right)}}\, dx = C + 4 \int \frac{x^{2}}{\operatorname{atan}{\left(2 x \right)}}\, dx + \int \frac{1}{\operatorname{atan}{\left(2 x \right)}}\, dx$$

Simplify

The answer [src]

  1             
  /             
 |              
 |          2   
 |   1 + 4*x    
 |  --------- dx
 |  atan(2*x)   
 |              
/               
0

$$\int\limits_{0}^{1} \frac{4 x^{2} + 1}{\operatorname{atan}{\left(2 x \right)}}\, dx$$

  1             
  /             
 |              
 |          2   
 |   1 + 4*x    
 |  --------- dx
 |  atan(2*x)   
 |              
/               
0

$$\int\limits_{0}^{1} \frac{4 x^{2} + 1}{\operatorname{atan}{\left(2 x \right)}}\, dx$$

Integral((1 + 4*x^2)/atan(2*x), (x, 0, 1))

Numerical answer [src]

23.750004414302

23.750004414302

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

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Identical expressions

Similar expressions

Integral of 1/arctg2x(1+4x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution