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Integral of ln(t)^3 dt

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  x           
  /           
 |            
 |     3      
 |  log (t) dt
 |            
/             
2             
$$\int\limits_{2}^{x} \log{\left(t \right)}^{3}\, dt$$
Integral(log(t)^3, (t, 2, x))
Detail solution
  1. Let .

    Then let and substitute :

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. The integral of the exponential function is itself.

      Now evaluate the sub-integral.

    2. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. The integral of the exponential function is itself.

      Now evaluate the sub-integral.

    3. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. The integral of the exponential function is itself.

      Now evaluate the sub-integral.

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

      1. The integral of the exponential function is itself.

      So, the result is:

    Now substitute back in:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                           
 |                                                            
 |    3                        3             2                
 | log (t) dt = C - 6*t + t*log (t) - 3*t*log (t) + 6*t*log(t)
 |                                                            
/                                                             
$$\int \log{\left(t \right)}^{3}\, dt = C + t \log{\left(t \right)}^{3} - 3 t \log{\left(t \right)}^{2} + 6 t \log{\left(t \right)} - 6 t$$
The answer [src]
                            3           2           3             2                
12 - 12*log(2) - 6*x - 2*log (2) + 6*log (2) + x*log (x) - 3*x*log (x) + 6*x*log(x)
$$x \log{\left(x \right)}^{3} - 3 x \log{\left(x \right)}^{2} + 6 x \log{\left(x \right)} - 6 x - 12 \log{\left(2 \right)} - 2 \log{\left(2 \right)}^{3} + 6 \log{\left(2 \right)}^{2} + 12$$
=
=
                            3           2           3             2                
12 - 12*log(2) - 6*x - 2*log (2) + 6*log (2) + x*log (x) - 3*x*log (x) + 6*x*log(x)
$$x \log{\left(x \right)}^{3} - 3 x \log{\left(x \right)}^{2} + 6 x \log{\left(x \right)} - 6 x - 12 \log{\left(2 \right)} - 2 \log{\left(2 \right)}^{3} + 6 \log{\left(2 \right)}^{2} + 12$$

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