Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of e^x/(1+e^x)
Integral of sin^2(3x)
Integral of sin(x)/cos(x)
Integral of |-1+x|
Identical expressions
(seven + two /sin(3x)+(five)/(four -x^ two))
(7 plus 2 divide by sinus of (3x) plus (5) divide by (4 minus x squared ))
(seven plus two divide by sinus of (3x) plus (five) divide by (four minus x to the power of two))
(7+2/sin(3x)+(5)/(4-x2))
7+2/sin3x+5/4-x2
(7+2/sin(3x)+(5)/(4-x²))
(7+2/sin(3x)+(5)/(4-x to the power of 2))
7+2/sin3x+5/4-x^2
(7+2 divide by sin(3x)+(5) divide by (4-x^2))
(7+2/sin(3x)+(5)/(4-x^2))dx
Similar expressions
(7-2/sin(3x)+(5)/(4-x^2))
(7+2/sin(3x)-(5)/(4-x^2))
(7+2/sin(3x)+(5)/(4+x^2))

Solving integrals/ (7+2/sin(3x)+(5)/(4-x^2))

Integral of (7+2/sin(3x)+(5)/(4-x^2)) dx

The solution

You have entered [src]

  1                           
  /                           
 |                            
 |  /       2         5   \   
 |  |7 + -------- + ------| dx
 |  |    sin(3*x)        2|   
 |  \               4 - x /   
 |                            
/                             
0

$$\int\limits_{0}^{1} \left(7 + \frac{2}{\sin{\left(3 x \right)}} + \frac{5}{4 - x^{2}}\right)\, dx$$

Integral(7 + 2/sin(3*x) + 5/(4 - x^2), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 7\, dx = 7 x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2}{\sin{\left(3 x \right)}}\, dx = 2 \int \frac{1}{\sin{\left(3 x \right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{\log{\left(\cos{\left(3 x \right)} - 1 \right)}}{6} - \frac{\log{\left(\cos{\left(3 x \right)} + 1 \right)}}{6}$
  So, the result is: $\frac{\log{\left(\cos{\left(3 x \right)} - 1 \right)}}{3} - \frac{\log{\left(\cos{\left(3 x \right)} + 1 \right)}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{5}{4 - x^{2}}\, dx = 5 \int \frac{1}{4 - x^{2}}\, dx$
  1. Rewrite the integrand:
    
    $\frac{1}{4 - x^{2}} = - \frac{- \frac{1}{x + 2} + \frac{1}{x - 2}}{4}$
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{- \frac{1}{x + 2} + \frac{1}{x - 2}}{4}\right)\, dx = - \frac{\int \left(- \frac{1}{x + 2} + \frac{1}{x - 2}\right)\, dx}{4}$
    1. Integrate term-by-term:
      1. The integral of $\frac{1}{x - 2}$ is $\log{\left(x - 2 \right)}$ .
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \left(- \frac{1}{x + 2}\right)\, dx = - \int \frac{1}{x + 2}\, dx$
        
        The integral of $\frac{1}{x + 2}$ is $\log{\left(x + 2 \right)}$ .
        
        So, the result is: $- \log{\left(x + 2 \right)}$
      The result is: $\log{\left(x - 2 \right)} - \log{\left(x + 2 \right)}$
    So, the result is: $- \frac{\log{\left(x - 2 \right)}}{4} + \frac{\log{\left(x + 2 \right)}}{4}$
  So, the result is: $- \frac{5 \log{\left(x - 2 \right)}}{4} + \frac{5 \log{\left(x + 2 \right)}}{4}$
The result is: $7 x - \frac{5 \log{\left(x - 2 \right)}}{4} + \frac{5 \log{\left(x + 2 \right)}}{4} + \frac{\log{\left(\cos{\left(3 x \right)} - 1 \right)}}{3} - \frac{\log{\left(\cos{\left(3 x \right)} + 1 \right)}}{3}$
Add the constant of integration:

$7 x - \frac{5 \log{\left(x - 2 \right)}}{4} + \frac{5 \log{\left(x + 2 \right)}}{4} + \frac{\log{\left(\cos{\left(3 x \right)} - 1 \right)}}{3} - \frac{\log{\left(\cos{\left(3 x \right)} + 1 \right)}}{3}+ \mathrm{constant}$

The answer is:

$7 x - \frac{5 \log{\left(x - 2 \right)}}{4} + \frac{5 \log{\left(x + 2 \right)}}{4} + \frac{\log{\left(\cos{\left(3 x \right)} - 1 \right)}}{3} - \frac{\log{\left(\cos{\left(3 x \right)} + 1 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                            
 |                                                                                                             
 | /       2         5   \                5*log(-2 + x)   log(1 + cos(3*x))   log(-1 + cos(3*x))   5*log(2 + x)
 | |7 + -------- + ------| dx = C + 7*x - ------------- - ----------------- + ------------------ + ------------
 | |    sin(3*x)        2|                      4                 3                   3                 4      
 | \               4 - x /                                                                                     
 |                                                                                                             
/

$$\int \left(7 + \frac{2}{\sin{\left(3 x \right)}} + \frac{5}{4 - x^{2}}\right)\, dx = C + 7 x - \frac{5 \log{\left(x - 2 \right)}}{4} + \frac{5 \log{\left(x + 2 \right)}}{4} + \frac{\log{\left(\cos{\left(3 x \right)} - 1 \right)}}{3} - \frac{\log{\left(\cos{\left(3 x \right)} + 1 \right)}}{3}$$

Simplify

The answer [src]

     11*pi*I
oo - -------
        12

$$\infty - \frac{11 i \pi}{12}$$

     11*pi*I
oo - -------
        12

$$\infty - \frac{11 i \pi}{12}$$

Упростить

Numerical answer [src]

39.2607697099495

39.2607697099495

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (7+2/sin(3x)+(5)/(4-x^2)) dx

Limits of integration:

The graph:

Piecewise:

The solution