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Identical expressions
one /(x^ two + nine)^ three
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Similar expressions
1/(x^2-9)^3

Solving integrals/ 1/(x^2+9)^3

Integral of 1/(x^2+9)^3 dx

The solution

You have entered [src]

  1               
  /               
 |                
 |        1       
 |  1*--------- dx
 |            3   
 |    / 2    \    
 |    \x  + 9/    
 |                
/                 
0

$$\int\limits_{0}^{1} 1 \cdot \frac{1}{\left(x^{2} + 9\right)^{3}}\, dx$$

Integral(1/(x^2 + 9)^3, (x, 0, 1))

Detail solution

TrigSubstitutionRule(theta=_theta, func=3*tan(_theta), rewritten=cos(_theta)**4/243, substep=ConstantTimesRule(constant=1/243, other=cos(_theta)**4, substep=RewriteRule(rewritten=(cos(2*_theta)/2 + 1/2)**2, substep=AlternativeRule(alternatives=[RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u)/4, symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), 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)/2, symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u)/4, symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), 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)/2, symbol=_u), context=cos(2*_theta), symbol=_theta), context=cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 + cos(2*_theta)/2 + 1/4, symbol=_theta), context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta)], context=(cos(2*_theta)/2 + 1/2)**2, symbol=_theta), context=cos(_theta)**4, symbol=_theta), context=cos(_theta)**4/243, symbol=_theta), restriction=True, context=1/(x**2 + 9)**3, symbol=x)

Now simplify:

$\frac{x^{3}}{216 \left(x^{2} + 9\right)^{2}} + \frac{5 x}{72 \left(x^{2} + 9\right)^{2}} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{648}$
Add the constant of integration:

$\frac{x^{3}}{216 \left(x^{2} + 9\right)^{2}} + \frac{5 x}{72 \left(x^{2} + 9\right)^{2}} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{648}+ \mathrm{constant}$

The answer is:

$\frac{x^{3}}{216 \left(x^{2} + 9\right)^{2}} + \frac{5 x}{72 \left(x^{2} + 9\right)^{2}} + \frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{648}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                         /x\                               
 |                      atan|-|                      /     2\ 
 |       1                  \3/        x           x*\9 - x / 
 | 1*--------- dx = C + ------- + ------------ + -------------
 |           3            648         /     2\               2
 |   / 2    \                     162*\9 + x /       /     2\ 
 |   \x  + 9/                                    648*\9 + x / 
 |                                                            
/

$${{x^3+15\,x}\over{216\,x^4+3888\,x^2+17496}}+{{\arctan \left({{x }\over{3}}\right)}\over{648}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

 1     atan(1/3)
---- + ---------
1350      648

$${{25\,\arctan \left({{1}\over{3}}\right)+12}\over{16200}}$$

 1     atan(1/3)
---- + ---------
1350      648

$$\frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{648} + \frac{1}{1350}$$

Simplify

Numerical answer [src]

0.00123726937406889

0.00123726937406889

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^2+9)^3 dx

Limits of integration:

The graph:

Piecewise:

The solution