Integral of cosx+3x-lnx dx

You have entered [src]

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

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

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

Detail solution

Integrate term-by-term:
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 x\, dx = 3 \int x\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $\frac{3 x^{2}}{2}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  The result is: $\frac{3 x^{2}}{2} + \sin{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \log{\left(x \right)}\right)\, dx = - \int \log{\left(x \right)}\, dx$
  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:
    
    $\int 1\, dx = x$
  So, the result is: $- x \log{\left(x \right)} + x$
The result is: $\frac{3 x^{2}}{2} - x \log{\left(x \right)} + x + \sin{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

5/2 + sin(1)

$$\sin{\left(1 \right)} + \frac{5}{2}$$

=

5/2 + sin(1)

$$\sin{\left(1 \right)} + \frac{5}{2}$$

5/2 + sin(1)

Numerical answer [src]

3.3414709848079

3.3414709848079

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of cosx+3x-lnx dx

Limits of integration:

The graph:

Piecewise:

The solution