Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of dx/x(1+lnx)
Integral of sin^3xdx
Integral of (3sinx-2cosx)/(1+cosx)
Integral of cos^3xsin^2x
Identical expressions
(3sinx-2cosx)/(one +cosx)
(3 sinus of x minus 2 co sinus of e of x) divide by (1 plus co sinus of e of x)
(3 sinus of x minus 2 co sinus of e of x) divide by (one plus co sinus of e of x)
3sinx-2cosx/1+cosx
(3sinx-2cosx) divide by (1+cosx)
(3sinx-2cosx)/(1+cosx)dx
Similar expressions
(3sinx-2cosx)/(1-cosx)
(3sinx+2cosx)/(1+cosx)

Solving integrals/ (3sinx-2cosx)/(1+cosx)

Integral of (3sinx-2cosx)/(1+cosx) dx

The solution

You have entered [src]

  1                       
  /                       
 |                        
 |  3*sin(x) - 2*cos(x)   
 |  ------------------- dx
 |       1 + cos(x)       
 |                        
/                         
0

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

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

Detail solution

Rewrite the integrand:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$$-4\,\left(\arctan \left({{\sin x}\over{\cos x+1}}\right)-{{\sin x }\over{2\,\left(\cos x+1\right)}}\right)-3\,\log \left(\cos x+1 \right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                       /       2     \
-2 + 2*tan(1/2) + 3*log\1 + tan (1/2)/

$$-{{4\,\cos 1\,\arctan \left({{\sin 1}\over{\cos 1+1}}\right)}\over{ \cos 1+1}}-{{4\,\arctan \left({{\sin 1}\over{\cos 1+1}}\right) }\over{\cos 1+1}}-3\,\log \left(\cos 1+1\right)+3\,\log 2+{{2\,\sin 1}\over{\cos 1+1}}$$

                       /       2     \
-2 + 2*tan(1/2) + 3*log\1 + tan (1/2)/

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

Simplify

Numerical answer [src]

-0.123889577650083

-0.123889577650083

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (3sinx-2cosx)/(1+cosx) dx

Limits of integration:

The graph:

Piecewise:

The solution