Mister Exam

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

Solving integrals/ -((1/cosx)+(1/(3cosx)))

Integral of -((1/cosx)+(1/(3cosx))) dx

The solution

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

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

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

Detail solution

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

$\frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{3} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{3} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  2*log(1 + sin(1))   2*log(1 - sin(1))
- ----------------- + -----------------
          3                   3

$$\frac{2 \log{\left(1 - \sin{\left(1 \right)} \right)}}{3} - \frac{2 \log{\left(\sin{\left(1 \right)} + 1 \right)}}{3}$$

  2*log(1 + sin(1))   2*log(1 - sin(1))
- ----------------- + -----------------
          3                   3

$$\frac{2 \log{\left(1 - \sin{\left(1 \right)} \right)}}{3} - \frac{2 \log{\left(\sin{\left(1 \right)} + 1 \right)}}{3}$$

-2*log(1 + sin(1))/3 + 2*log(1 - sin(1))/3

Numerical answer [src]

-1.63492156117802

-1.63492156117802

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

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Integral of -((1/cosx)+(1/(3cosx))) dx

Limits of integration:

The graph:

Piecewise:

The solution