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Identical expressions
x(tan^ two (x))
x( tangent of squared (x))
x( tangent of to the power of two (x))
x(tan2(x))
xtan2x
x(tan²(x))
x(tan to the power of 2(x))
xtan^2x
x(tan^2(x))dx

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

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

The solution

You have entered [src]

  1             
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 |  x*tan (x) dx
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0

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

Integral(x*tan(x)^2, (x, 0, 1))

Detail solution

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.
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: $- \frac{x^{2}}{2} - \log{\left(\cos{\left(x \right)} \right)}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

         /       2   \         
  1   log\1 + tan (1)/         
- - - ---------------- + tan(1)
  2          2

$${{\left(\sin ^22+\cos ^22+2\,\cos 2+1\right)\,\log \left(\sin ^22+ \cos ^22+2\,\cos 2+1\right)-\sin ^22+4\,\sin 2-\cos ^22-2\,\cos 2-1 }\over{2\,\sin ^22+2\,\cos ^22+4\,\cos 2+2}}-{{\log 4}\over{2}}$$

         /       2   \         
  1   log\1 + tan (1)/         
- - - ---------------- + tan(1)
  2          2

$$- \frac{\log{\left(1 + \tan^{2}{\left(1 \right)} \right)}}{2} - \frac{1}{2} + \tan{\left(1 \right)}$$

Simplify

Numerical answer [src]

0.441781254268888

0.441781254268888

The graph

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

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How to use it?

Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution