Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x*cos(x/2)
Integral of y^(-2/3)
Integral of (tan(x))^2/(cos(x))^2
Integral of tan2xdx
Identical expressions
sin(2x)*sin(x)/(one +cos(x))
sinus of (2x) multiply by sinus of (x) divide by (1 plus co sinus of e of (x))
sinus of (2x) multiply by sinus of (x) divide by (one plus co sinus of e of (x))
sin(2x)sin(x)/(1+cos(x))
sin2xsinx/1+cosx
sin(2x)*sin(x) divide by (1+cos(x))
sin(2x)*sin(x)/(1+cos(x))dx
Similar expressions
sin(2x)*sin(x)/(1-cos(x))
sin(2x)*sinx/(1+cosx)

Solving integrals/ sin(2x)*sin(x)/(1+cos(x))

Integral of sin(2x)*sin(x)/(1+cos(x)) dx

The solution

You have entered [src]

  1                   
  /                   
 |                    
 |  sin(2*x)*sin(x)   
 |  --------------- dx
 |     1 + cos(x)     
 |                    
/                     
0

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

Integral((sin(2*x)*sin(x))/(1 + cos(x)), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}\, dx = 2 \int \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $- \frac{x \tan^{4}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{2 x \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{x}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{6 \tan^{3}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2}$
  So, the result is: $- \frac{2 x \tan^{4}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{4 x \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{2 x}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{12 \tan^{3}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{4 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2}$
Method #2
1. Rewrite the integrand:
  
  $\frac{\sin{\left(x \right)} \sin{\left(2 x \right)}}{\cos{\left(x \right)} + 1} = \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}\, dx = 2 \int \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $- \frac{x \tan^{4}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{2 x \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{x}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{6 \tan^{3}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{2 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2}$
  So, the result is: $- \frac{2 x \tan^{4}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{4 x \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{2 x}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{12 \tan^{3}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{4 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2}$
Now simplify:

$- x + 2 \sin{\left(x \right)} - \frac{\sin{\left(2 x \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                                   /x\                         3/x\                        2/x\                        4/x\       
 |                                                               4*tan|-|                   12*tan |-|                 4*x*tan |-|                 2*x*tan |-|       
 | sin(2*x)*sin(x)                     2*x                            \2/                          \2/                         \2/                         \2/       
 | --------------- dx = C - ------------------------- + ------------------------- + ------------------------- - ------------------------- - -------------------------
 |    1 + cos(x)                     4/x\        2/x\            4/x\        2/x\            4/x\        2/x\            4/x\        2/x\            4/x\        2/x\
 |                          2 + 2*tan |-| + 4*tan |-|   2 + 2*tan |-| + 4*tan |-|   2 + 2*tan |-| + 4*tan |-|   2 + 2*tan |-| + 4*tan |-|   2 + 2*tan |-| + 4*tan |-|
/                                     \2/         \2/             \2/         \2/             \2/         \2/             \2/         \2/             \2/         \2/

$$\int \frac{\sin{\left(x \right)} \sin{\left(2 x \right)}}{\cos{\left(x \right)} + 1}\, dx = C - \frac{2 x \tan^{4}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{4 x \tan^{2}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} - \frac{2 x}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{12 \tan^{3}{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2} + \frac{4 \tan{\left(\frac{x}{2} \right)}}{2 \tan^{4}{\left(\frac{x}{2} \right)} + 4 \tan^{2}{\left(\frac{x}{2} \right)} + 2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                                2                               4                                                                3             
                2                          4*tan (1/2)                     2*tan (1/2)                      4*tan(1/2)                     12*tan (1/2)        
- ----------------------------- - ----------------------------- - ----------------------------- + ----------------------------- + -----------------------------
           4             2                 4             2                 4             2                 4             2                 4             2     
  2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)

$$- \frac{2}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} - \frac{4 \tan^{2}{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} - \frac{2 \tan^{4}{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} + \frac{12 \tan^{3}{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2} + \frac{4 \tan{\left(\frac{1}{2} \right)}}{2 \tan^{4}{\left(\frac{1}{2} \right)} + 4 \tan^{2}{\left(\frac{1}{2} \right)} + 2}$$

                                                2                               4                                                                3             
                2                          4*tan (1/2)                     2*tan (1/2)                      4*tan(1/2)                     12*tan (1/2)        
- ----------------------------- - ----------------------------- - ----------------------------- + ----------------------------- + -----------------------------
           4             2                 4             2                 4             2                 4             2                 4             2     
  2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)   2 + 2*tan (1/2) + 4*tan (1/2)

-2/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) - 4*tan(1/2)^2/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) - 2*tan(1/2)^4/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) + 4*tan(1/2)/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2) + 12*tan(1/2)^3/(2 + 2*tan(1/2)^4 + 4*tan(1/2)^2)

Numerical answer [src]

0.228293256202952

0.228293256202952

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sin(2x)*sin(x)/(1+cos(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2