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Similar expressions
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Solving integrals/ 1/(x^4+2x^2+1)

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

The solution

You have entered [src]

  1                 
  /                 
 |                  
 |        1         
 |  ------------- dx
 |   4      2       
 |  x  + 2*x  + 1   
 |                  
/                   
0

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

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

Detail solution

Rewrite the integrand:

$\frac{1}{\left(x^{4} + 2 x^{2}\right) + 1} = \frac{1}{\left(x^{2} + 1\right)^{2}}$

TrigSubstitutionRule(theta=_theta, func=tan(_theta), rewritten=cos(_theta)**2, substep=RewriteRule(rewritten=cos(2*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(2*_theta)/2 + 1/2, symbol=_theta), context=cos(_theta)**2, symbol=_theta), restriction=True, context=(x**2 + 1)**(-2), symbol=x)
Now simplify:

$\frac{x + \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}{2 \left(x^{2} + 1\right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   pi
- + --
4   8

$$\frac{1}{4} + \frac{\pi}{8}$$

1   pi
- + --
4   8

$$\frac{1}{4} + \frac{\pi}{8}$$

1/4 + pi/8

Numerical answer [src]

0.642699081698724

0.642699081698724

The graph

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution