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Integral of d{x}:
Integral of cosx^2
Integral of sinh(x)
Integral of 2/x^4
Integral of 1/(1+4x^2)
Identical expressions
(x^ two + three x- one)/cos(x/3)
(x squared plus 3x minus 1) divide by co sinus of e of (x divide by 3)
(x to the power of two plus three x minus one) divide by co sinus of e of (x divide by 3)
(x2+3x-1)/cos(x/3)
x2+3x-1/cosx/3
(x²+3x-1)/cos(x/3)
(x to the power of 2+3x-1)/cos(x/3)
x^2+3x-1/cosx/3
(x^2+3x-1) divide by cos(x divide by 3)
(x^2+3x-1)/cos(x/3)dx
Similar expressions
(x^2-3x-1)/cos(x/3)
(x^2+3x+1)/cos(x/3)

Integral of (x^2+3x-1)/cos(x/3) dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |   2             
 |  x  + 3*x - 1   
 |  ------------ dx
 |        /x\      
 |     cos|-|      
 |        \3/      
 |                 
/                  
0

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

Integral((x^2 + 3*x - 1)/cos(x/3), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{\left(x^{2} + 3 x\right) - 1}{\cos{\left(\frac{x}{3} \right)}} = \frac{x^{2}}{\cos{\left(\frac{x}{3} \right)}} + \frac{3 x}{\cos{\left(\frac{x}{3} \right)}} - \frac{1}{\cos{\left(\frac{x}{3} \right)}}$
Integrate term-by-term:
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\int \frac{x^{2}}{\cos{\left(\frac{x}{3} \right)}}\, dx$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3 x}{\cos{\left(\frac{x}{3} \right)}}\, dx = 3 \int \frac{x}{\cos{\left(\frac{x}{3} \right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\int \frac{x}{\cos{\left(\frac{x}{3} \right)}}\, dx$
  So, the result is: $3 \int \frac{x}{\cos{\left(\frac{x}{3} \right)}}\, dx$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{\cos{\left(\frac{x}{3} \right)}}\right)\, dx = - \int \frac{1}{\cos{\left(\frac{x}{3} \right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $- \frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} - 1 \right)}}{2} + \frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} + 1 \right)}}{2}$
  So, the result is: $\frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} - 1 \right)}}{2} - \frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} + 1 \right)}}{2}$
The result is: $\frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} - 1 \right)}}{2} - \frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} + 1 \right)}}{2} + 3 \int \frac{x}{\cos{\left(\frac{x}{3} \right)}}\, dx + \int \frac{x^{2}}{\cos{\left(\frac{x}{3} \right)}}\, dx$
Add the constant of integration:

$\frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} - 1 \right)}}{2} - \frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} + 1 \right)}}{2} + 3 \int \frac{x}{\cos{\left(\frac{x}{3} \right)}}\, dx + \int \frac{x^{2}}{\cos{\left(\frac{x}{3} \right)}}\, dx+ \mathrm{constant}$

The answer is:

$\frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} - 1 \right)}}{2} - \frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} + 1 \right)}}{2} + 3 \int \frac{x}{\cos{\left(\frac{x}{3} \right)}}\, dx + \int \frac{x^{2}}{\cos{\left(\frac{x}{3} \right)}}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                  /         
 |                           /                 /       /x\\        /        /x\\    |          
 |  2                       |             3*log|1 + sin|-||   3*log|-1 + sin|-||    |    2     
 | x  + 3*x - 1             |   x              \       \3//        \        \3//    |   x      
 | ------------ dx = C + 3* | ------ dx - ----------------- + ------------------ +  | ------ dx
 |       /x\                |    /x\              2                   2             |    /x\   
 |    cos|-|                | cos|-|                                                | cos|-|   
 |       \3/                |    \3/                                                |    \3/   
 |                          |                                                       |          
/                          /                                                       /

$$\int \frac{\left(x^{2} + 3 x\right) - 1}{\cos{\left(\frac{x}{3} \right)}}\, dx = C + \frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} - 1 \right)}}{2} - \frac{3 \log{\left(\sin{\left(\frac{x}{3} \right)} + 1 \right)}}{2} + 3 \int \frac{x}{\cos{\left(\frac{x}{3} \right)}}\, dx + \int \frac{x^{2}}{\cos{\left(\frac{x}{3} \right)}}\, dx$$

Simplify

The answer [src]

  1                 
  /                 
 |                  
 |        2         
 |  -1 + x  + 3*x   
 |  ------------- dx
 |         /x\      
 |      cos|-|      
 |         \3/      
 |                  
/                   
0

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

  1                 
  /                 
 |                  
 |        2         
 |  -1 + x  + 3*x   
 |  ------------- dx
 |         /x\      
 |      cos|-|      
 |         \3/      
 |                  
/                   
0

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

Integral((-1 + x^2 + 3*x)/cos(x/3), (x, 0, 1))

Numerical answer [src]

0.868773043201666

0.868773043201666

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

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Identical expressions

Similar expressions

Integral of (x^2+3x-1)/cos(x/3) dx

Limits of integration:

The graph:

Piecewise:

The solution