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

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

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The solution

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∫01xtan⁡2(x) dx\int\limits_{0}^{1} x \tan^{2}{\left(x \right)}\, dx 
Integral(x*tan(x)^2, (x, 0, 1))
Detail solution

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=xu{\left(x \right)} = x and let dv⁡(x)=tan⁡2(x)\operatorname{dv}{\left(x \right)} = \tan^{2}{\left(x \right)}.

    Then du⁡(x)=1\operatorname{du}{\left(x \right)} = 1.

    To find v(x)v{\left(x \right)}:

    1. Rewrite the integrand:

      tan⁡2(x)=sec⁡2(x)−1\tan^{2}{\left(x \right)} = \sec^{2}{\left(x \right)} - 1

    2. Integrate term-by-term:

      1. ∫sec⁡2(x) dx=tan⁡(x)\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:

        ∫(−1) dx=−x\int \left(-1\right)\, dx = - x

      The result is: −x+tan⁡(x)- 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:

      ∫(−x) dx=−∫x dx\int \left(- x\right)\, dx = - \int x\, dx

      1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫x dx=x22\int x\, dx = \frac{x^{2}}{2}

      So, the result is: −x22- \frac{x^{2}}{2}

    1. Rewrite the integrand:

      tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

    2. Let u=cos⁡(x)u = \cos{\left(x \right)}.

      Then let du=−sin⁡(x)dxdu = - \sin{\left(x \right)} dx and substitute −du- du:

      ∫1u du\int \frac{1}{u}\, du

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

        ∫(−1u) du=−∫1u du\int \left(- \frac{1}{u}\right)\, du = - \int \frac{1}{u}\, du

        1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

        So, the result is: −log⁡(u)- \log{\left(u \right)}

      Now substitute uu back in:

      −log⁡(cos⁡(x))- \log{\left(\cos{\left(x \right)} \right)}

    The result is: −x22−log⁡(cos⁡(x))- \frac{x^{2}}{2} - \log{\left(\cos{\left(x \right)} \right)}

  3. Now simplify:

    −x22+xtan⁡(x)+log⁡(cos⁡(x))- \frac{x^{2}}{2} + x \tan{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)}

  4. Add the constant of integration:

    −x22+xtan⁡(x)+log⁡(cos⁡(x))+constant- \frac{x^{2}}{2} + x \tan{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)}+ \mathrm{constant}

The answer is:

−x22+xtan⁡(x)+log⁡(cos⁡(x))+constant- \frac{x^{2}}{2} + x \tan{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                     
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 | x*tan (x) dx = C + -- + x*(-x + tan(x)) + log(cos(x))
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(sin⁡2(2 x)+cos⁡2(2 x)+2 cos⁡(2 x)+1) log⁡(sin⁡2(2 x)+cos⁡2(2 x)+2 cos⁡(2 x)+1)−x2 sin⁡2(2 x)+4 x sin⁡(2 x)−x2 cos⁡2(2 x)−2 x2 cos⁡(2 x)−x22 sin⁡2(2 x)+2 cos⁡2(2 x)+4 cos⁡(2 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
0.001.000.100.200.300.400.500.600.700.800.900.02.5
Plot the graph f(x)
The answer [src] 
         /       2   \         
  1   log\1 + tan (1)/         
- - - ---------------- + tan(1)
  2          2
(sin⁡22+cos⁡22+2 cos⁡2+1) log⁡(sin⁡22+cos⁡22+2 cos⁡2+1)−sin⁡22+4 sin⁡2−cos⁡22−2 cos⁡2−12 sin⁡22+2 cos⁡22+4 cos⁡2+2−log⁡42{{\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
−log⁡(1+tan⁡2(1))2−12+tan⁡(1)- \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
Integral of x(tan^2(x)) dx

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

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