Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of sin(5x)
Integral of 0dx
Integral of -x+4
Integral of (x-2)^2dx
Identical expressions
y= two (lnx-cos3x)
y equally 2(lnx minus co sinus of e of 3x)
y equally two (lnx minus co sinus of e of 3x)
y=2lnx-cos3x
y=2(lnx-cos3x)dx
Similar expressions
y=2(lnx+cos3x)

Integral of y=2(lnx-cos3x) dx

The solution

You have entered [src]

  1                         
  /                         
 |                          
 |  2*(log(x) - cos(3*x)) dx
 |                          
/                           
0

$$\int\limits_{0}^{1} 2 \left(\log{\left(x \right)} - \cos{\left(3 x \right)}\right)\, dx$$

Integral(2*(log(x) - cos(3*x)), (x, 0, 1))

Detail solution

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

$\int 2 \left(\log{\left(x \right)} - \cos{\left(3 x \right)}\right)\, dx = 2 \int \left(\log{\left(x \right)} - \cos{\left(3 x \right)}\right)\, dx$
1. Integrate term-by-term:
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = \log{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
    
    Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .
    
    To find $v{\left(x \right)}$ :
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
    Now evaluate the sub-integral.
  2. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \cos{\left(3 x \right)}\right)\, dx = - \int \cos{\left(3 x \right)}\, dx$
    1. Let $u = 3 x$ .
      
      Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
      
      $\int \frac{\cos{\left(u \right)}}{9}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{\cos{\left(u \right)}}{3}\, du = \frac{\int \cos{\left(u \right)}\, du}{3}$
        
        The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
        
        So, the result is: $\frac{\sin{\left(u \right)}}{3}$
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(3 x \right)}}{3}$
    So, the result is: $- \frac{\sin{\left(3 x \right)}}{3}$
  The result is: $x \log{\left(x \right)} - x - \frac{\sin{\left(3 x \right)}}{3}$
So, the result is: $2 x \log{\left(x \right)} - 2 x - \frac{2 \sin{\left(3 x \right)}}{3}$
Add the constant of integration:

$2 x \log{\left(x \right)} - 2 x - \frac{2 \sin{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer is:

$2 x \log{\left(x \right)} - 2 x - \frac{2 \sin{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                            
 |                                      2*sin(3*x)             
 | 2*(log(x) - cos(3*x)) dx = C - 2*x - ---------- + 2*x*log(x)
 |                                          3                  
/

$$\int 2 \left(\log{\left(x \right)} - \cos{\left(3 x \right)}\right)\, dx = C + 2 x \log{\left(x \right)} - 2 x - \frac{2 \sin{\left(3 x \right)}}{3}$$

Simplify

The answer [src]

     2*sin(3)
-2 - --------
        3

$$-2 - \frac{2 \sin{\left(3 \right)}}{3}$$

     2*sin(3)
-2 - --------
        3

$$-2 - \frac{2 \sin{\left(3 \right)}}{3}$$

Упростить

Numerical answer [src]

-2.09408000537324

-2.09408000537324

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of y=2(lnx-cos3x) dx

Limits of integration:

The graph:

Piecewise:

The solution