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

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

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∫π4π34xtan⁡2(x) dx\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

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

    ∫4xtan⁡2(x) dx=4∫xtan⁡2(x) dx\int 4 x \tan^{2}{\left(x \right)}\, dx = 4 \int x \tan^{2}{\left(x \right)}\, dx

    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)}

    So, the result is: 2x2+4x(−x+tan⁡(x))+4log⁡(cos⁡(x))2 x^{2} + 4 x \left(- x + \tan{\left(x \right)}\right) + 4 \log{\left(\cos{\left(x \right)} \right)}

  2. Now simplify:

    −2x2+4xtan⁡(x)+4log⁡(cos⁡(x))- 2 x^{2} + 4 x \tan{\left(x \right)} + 4 \log{\left(\cos{\left(x \right)} \right)}

  3. Add the constant of integration:

    −2x2+4xtan⁡(x)+4log⁡(cos⁡(x))+constant- 2 x^{2} + 4 x \tan{\left(x \right)} + 4 \log{\left(\cos{\left(x \right)} \right)}+ \mathrm{constant}

The answer is:

−2x2+4xtan⁡(x)+4log⁡(cos⁡(x))+constant- 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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 | 4*x*tan (x) dx = C + 2*x  + 4*log(cos(x)) + 4*x*(-x + tan(x))
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4 ((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)−x2)2 sin⁡2(2 x)+2 cos⁡2(2 x)+4 cos⁡(2 x)+2{{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
0.8000.8250.8500.8750.9000.9250.9500.9751.0001.025020
Plot the graph f(x)
The answer [src] 
                                2          ___
                            7*pi    4*pi*\/ 3 
-pi - 2*log(4) + 2*log(2) - ----- + ----------
                              72        3
−π−2log⁡(4)−7π272+2log⁡(2)+43π3- \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
−π−2log⁡(4)−7π272+2log⁡(2)+43π3- \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
Integral of 4*x*tan(x)^2 dx

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

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