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Identical expressions
four *x*tan(x)^ two
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4*x*tan(x)2
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4xtan(x)^2
4xtan(x)2
4xtanx2
4xtanx^2
4*x*tan(x)^2dx

Solving integrals/ 4*x*tan(x)^2

Integral of 4xtan(x)^2 dx

The solution

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 pi               
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4

$$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{3}} 4 x \tan^{2}{\left(x \right)}\, dx$$

Integral(4*x*tan(x)^2, (x, pi/4, pi/3))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 4 x \tan^{2}{\left(x \right)}\, dx = 4 \int x \tan^{2}{\left(x \right)}\, dx$
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \tan^{2}{\left(x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 1$ .
  
  To find $v{\left(x \right)}$ :
  1. Rewrite the integrand:
    
    $\tan^{2}{\left(x \right)} = \sec^{2}{\left(x \right)} - 1$
  2. Integrate term-by-term:
    1. $\int \sec^{2}{\left(x \right)}\, dx = \tan{\left(x \right)}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int \left(-1\right)\, dx = - x$
    The result is: $- x + \tan{\left(x \right)}$
  Now evaluate the sub-integral.
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- x\right)\, dx = - \int x\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $- \frac{x^{2}}{2}$
  1. Rewrite the integrand:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  2. Let $u = \cos{\left(x \right)}$ .
    
    Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
    
    $\int \frac{1}{u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{u}\right)\, du = - \int \frac{1}{u}\, du$
      1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      So, the result is: $- \log{\left(u \right)}$
    Now substitute $u$ back in:
    
    $- \log{\left(\cos{\left(x \right)} \right)}$
  The result is: $- \frac{x^{2}}{2} - \log{\left(\cos{\left(x \right)} \right)}$
So, the result is: $2 x^{2} + 4 x \left(- x + \tan{\left(x \right)}\right) + 4 \log{\left(\cos{\left(x \right)} \right)}$
Now simplify:

$- 2 x^{2} + 4 x \tan{\left(x \right)} + 4 \log{\left(\cos{\left(x \right)} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                             
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 |        2                2                                    
 | 4*x*tan (x) dx = C + 2*x  + 4*log(cos(x)) + 4*x*(-x + tan(x))
 |                                                              
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                2          ___
                            7*pi    4*pi*\/ 3 
-pi - 2*log(4) + 2*log(2) - ----- + ----------
                              72        3

$$- \pi - 2 \log{\left(4 \right)} - \frac{7 \pi^{2}}{72} + 2 \log{\left(2 \right)} + \frac{4 \sqrt{3} \pi}{3}$$

                                2          ___
                            7*pi    4*pi*\/ 3 
-pi - 2*log(4) + 2*log(2) - ----- + ----------
                              72        3

$$- \pi - 2 \log{\left(4 \right)} - \frac{7 \pi^{2}}{72} + 2 \log{\left(2 \right)} + \frac{4 \sqrt{3} \pi}{3}$$

Simplify

Numerical answer [src]

1.76776556989906

1.76776556989906

The graph

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

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Identical expressions

Integral of 4xtan(x)^2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Integral of 4*x*tan(x)^2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of 4xtan(x)^2 dx