Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of 3
Integral of s
Integral of 1/(x^3)
Integral of tan
Identical expressions
sqrtx(lnx)dx
square root of x(lnx)dx
√x(lnx)dx
sqrtxlnxdx
Similar expressions
sqrt(x)(lnx)dx

Integral of sqrtx(lnx)dx dx

The solution

You have entered [src]

  3               
 e                
  /               
 |                
 |         2      
 |  t*x*log (x) dx
 |                
/                 
1

$$\int\limits_{1}^{e^{3}} t x \log{\left(x \right)}^{2}\, dx$$

Integral((t*x)*log(x)^2, (x, 1, exp(3)))

Detail solution

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

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

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

$t \left(\frac{x^{2} \log{\left(x \right)}^{2}}{2} - \frac{x^{2} \log{\left(x \right)}}{2} + \frac{x^{2}}{4}\right)$
Now simplify:

$\frac{t x^{2} \left(2 \log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 1\right)}{4}$
Add the constant of integration:

$\frac{t x^{2} \left(2 \log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 1\right)}{4}+ \mathrm{constant}$

The answer is:

$\frac{t x^{2} \left(2 \log{\left(x \right)}^{2} - 2 \log{\left(x \right)} + 1\right)}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                    
 |                        / 2    2    2       2       \
 |        2               |x    x *log (x)   x *log(x)|
 | t*x*log (x) dx = C + t*|-- + ---------- - ---------|
 |                        \4        2            2    /
/

$$\int t x \log{\left(x \right)}^{2}\, dx = C + t \left(\frac{x^{2} \log{\left(x \right)}^{2}}{2} - \frac{x^{2} \log{\left(x \right)}}{2} + \frac{x^{2}}{4}\right)$$

Simplify

The answer [src]

            6
  t   13*t*e 
- - + -------
  4      4

$$- \frac{t}{4} + \frac{13 t e^{6}}{4}$$

            6
  t   13*t*e 
- - + -------
  4      4

$$- \frac{t}{4} + \frac{13 t e^{6}}{4}$$

-t/4 + 13*t*exp(6)/4

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sqrtx(lnx)dx dx

Limits of integration:

The graph:

Piecewise:

The solution