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Integral of d{x}:
Integral of e^3x
Integral of sinx/cos2x
Integral of -e^x
Integral of sqrt(x^2+1)/x^3
Derivative of:
ln(t)^3
Identical expressions
ln(t)^ three
ln(t) cubed
ln(t) to the power of three
ln(t)3
lnt3
ln(t)³
ln(t) to the power of 3
lnt^3

Integral of ln(t)^3 dt

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

Let $u = \log{\left(t \right)}$ .

Then let $du = \frac{dt}{t}$ and substitute $du$ :

$\int u^{3} e^{u}\, du$
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(u \right)} = u^{3}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
  
  Then $\operatorname{du}{\left(u \right)} = 3 u^{2}$ .
  
  To find $v{\left(u \right)}$ :
  1. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  Now evaluate the sub-integral.
2. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(u \right)} = 3 u^{2}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
  
  Then $\operatorname{du}{\left(u \right)} = 6 u$ .
  
  To find $v{\left(u \right)}$ :
  1. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  Now evaluate the sub-integral.
3. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(u \right)} = 6 u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
  
  Then $\operatorname{du}{\left(u \right)} = 6$ .
  
  To find $v{\left(u \right)}$ :
  1. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  Now evaluate the sub-integral.
4. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 6 e^{u}\, du = 6 \int e^{u}\, du$
  1. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  So, the result is: $6 e^{u}$
Now substitute $u$ back in:

$t \log{\left(t \right)}^{3} - 3 t \log{\left(t \right)}^{2} + 6 t \log{\left(t \right)} - 6 t$
Now simplify:

$t \left(\log{\left(t \right)}^{3} - 3 \log{\left(t \right)}^{2} + 6 \log{\left(t \right)} - 6\right)$
Add the constant of integration:

$t \left(\log{\left(t \right)}^{3} - 3 \log{\left(t \right)}^{2} + 6 \log{\left(t \right)} - 6\right)+ \mathrm{constant}$

The answer is:

$t \left(\log{\left(t \right)}^{3} - 3 \log{\left(t \right)}^{2} + 6 \log{\left(t \right)} - 6\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                           
 |                                                            
 |    3                        3             2                
 | log (t) dt = C - 6*t + t*log (t) - 3*t*log (t) + 6*t*log(t)
 |                                                            
/

$$\int \log{\left(t \right)}^{3}\, dt = C + t \log{\left(t \right)}^{3} - 3 t \log{\left(t \right)}^{2} + 6 t \log{\left(t \right)} - 6 t$$

Simplify

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)

$$x \log{\left(x \right)}^{3} - 3 x \log{\left(x \right)}^{2} + 6 x \log{\left(x \right)} - 6 x - 12 \log{\left(2 \right)} - 2 \log{\left(2 \right)}^{3} + 6 \log{\left(2 \right)}^{2} + 12$$

Упростить

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

Other calculators

How to use it?

Identical expressions

Integral of ln(t)^3 dt

Limits of integration:

The graph:

Piecewise:

The solution