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Solving integrals/ sqrtx(lnx^2)

Integral of sqrtx(lnx^2) dx

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The solution

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

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

    ∫tx(log⁡(x)2)2 dx=t∫x(log⁡(x)2)2 dx\int t x \left(\log{\left(x \right)}^{2}\right)^{2}\, dx = t \int x \left(\log{\left(x \right)}^{2}\right)^{2}\, dx

    1. Don't know the steps in finding this integral.

      But the integral is

      x2log⁡(x)42−x2log⁡(x)3+3x2log⁡(x)22−3x2log⁡(x)2+3x24\frac{x^{2} \log{\left(x \right)}^{4}}{2} - x^{2} \log{\left(x \right)}^{3} + \frac{3 x^{2} \log{\left(x \right)}^{2}}{2} - \frac{3 x^{2} \log{\left(x \right)}}{2} + \frac{3 x^{2}}{4}

    So, the result is: t(x2log⁡(x)42−x2log⁡(x)3+3x2log⁡(x)22−3x2log⁡(x)2+3x24)t \left(\frac{x^{2} \log{\left(x \right)}^{4}}{2} - x^{2} \log{\left(x \right)}^{3} + \frac{3 x^{2} \log{\left(x \right)}^{2}}{2} - \frac{3 x^{2} \log{\left(x \right)}}{2} + \frac{3 x^{2}}{4}\right)

  2. Now simplify:

    tx2⋅(2log⁡(x)4−4log⁡(x)3+6log⁡(x)2−6log⁡(x)+3)4\frac{t x^{2} \cdot \left(2 \log{\left(x \right)}^{4} - 4 \log{\left(x \right)}^{3} + 6 \log{\left(x \right)}^{2} - 6 \log{\left(x \right)} + 3\right)}{4}

  3. Add the constant of integration:

    tx2⋅(2log⁡(x)4−4log⁡(x)3+6log⁡(x)2−6log⁡(x)+3)4+constant\frac{t x^{2} \cdot \left(2 \log{\left(x \right)}^{4} - 4 \log{\left(x \right)}^{3} + 6 \log{\left(x \right)}^{2} - 6 \log{\left(x \right)} + 3\right)}{4}+ \mathrm{constant}

The answer is:

tx2⋅(2log⁡(x)4−4log⁡(x)3+6log⁡(x)2−6log⁡(x)+3)4+constant\frac{t x^{2} \cdot \left(2 \log{\left(x \right)}^{4} - 4 \log{\left(x \right)}^{3} + 6 \log{\left(x \right)}^{2} - 6 \log{\left(x \right)} + 3\right)}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                                     
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 |            2            /   2    2    4                      2             2    2   \
 |        2                |3*x    x *log (x)    2    3      3*x *log(x)   3*x *log (x)|
 | t*x*log (x)  dx = C + t*|---- + ---------- - x *log (x) - ----------- + ------------|
 |                         \ 4         2                          2             2      /
/
∫tx(log⁡(x)2)2 dx=C+t(x2log⁡(x)42−x2log⁡(x)3+3x2log⁡(x)22−3x2log⁡(x)2+3x24)\int t x \left(\log{\left(x \right)}^{2}\right)^{2}\, dx = C + t \left(\frac{x^{2} \log{\left(x \right)}^{4}}{2} - x^{2} \log{\left(x \right)}^{3} + \frac{3 x^{2} \log{\left(x \right)}^{2}}{2} - \frac{3 x^{2} \log{\left(x \right)}}{2} + \frac{3 x^{2}}{4}\right)
Simplify
The answer [src] 
3*t
---
 4
3t4\frac{3 t}{4} 
=
= 
3*t
---
 4
3t4\frac{3 t}{4}
Упростить

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

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