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Integral of cosx/(3-2*sin^2x) dx

The solution

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

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

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

Detail solution

Rewrite the integrand:

$\frac{\cos{\left(x \right)}}{3 - 2 \sin^{2}{\left(x \right)}} = - \frac{\cos{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - 3}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{\cos{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - 3}\right)\, dx = - \int \frac{\cos{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - 3}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{\sqrt{6} \log{\left(\sin{\left(x \right)} - \frac{\sqrt{6}}{2} \right)}}{12} - \frac{\sqrt{6} \log{\left(\sin{\left(x \right)} + \frac{\sqrt{6}}{2} \right)}}{12}$
So, the result is: $- \frac{\sqrt{6} \log{\left(\sin{\left(x \right)} - \frac{\sqrt{6}}{2} \right)}}{12} + \frac{\sqrt{6} \log{\left(\sin{\left(x \right)} + \frac{\sqrt{6}}{2} \right)}}{12}$
Now simplify:

$\frac{\sqrt{6} \left(- \log{\left(\sin{\left(x \right)} - \frac{\sqrt{6}}{2} \right)} + \log{\left(\sin{\left(x \right)} + \frac{\sqrt{6}}{2} \right)}\right)}{12}$
Add the constant of integration:

$\frac{\sqrt{6} \left(- \log{\left(\sin{\left(x \right)} - \frac{\sqrt{6}}{2} \right)} + \log{\left(\sin{\left(x \right)} + \frac{\sqrt{6}}{2} \right)}\right)}{12}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{6} \left(- \log{\left(\sin{\left(x \right)} - \frac{\sqrt{6}}{2} \right)} + \log{\left(\sin{\left(x \right)} + \frac{\sqrt{6}}{2} \right)}\right)}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                   /    ___         \            /  ___         \
  /                         ___    |  \/ 6          |     ___    |\/ 6          |
 |                        \/ 6 *log|- ----- + sin(x)|   \/ 6 *log|----- + sin(x)|
 |     cos(x)                      \    2           /            \  2           /
 | ------------- dx = C - --------------------------- + -------------------------
 |          2                          12                           12           
 | 3 - 2*sin (x)                                                                 
 |                                                                               
/

$$\int \frac{\cos{\left(x \right)}}{3 - 2 \sin^{2}{\left(x \right)}}\, dx = C - \frac{\sqrt{6} \log{\left(\sin{\left(x \right)} - \frac{\sqrt{6}}{2} \right)}}{12} + \frac{\sqrt{6} \log{\left(\sin{\left(x \right)} + \frac{\sqrt{6}}{2} \right)}}{12}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

        /          /  ___         \\            /  ___\         /          /  ___\\            /  ___         \
    ___ |          |\/ 6          ||     ___    |\/ 6 |     ___ |          |\/ 6 ||     ___    |\/ 6          |
  \/ 6 *|pi*I + log|----- - sin(1)||   \/ 6 *log|-----|   \/ 6 *|pi*I + log|-----||   \/ 6 *log|----- + sin(1)|
        \          \  2           //            \  2  /         \          \  2  //            \  2           /
- ---------------------------------- - ---------------- + ------------------------- + -------------------------
                  12                          12                      12                          12

$$- \frac{\sqrt{6} \log{\left(\frac{\sqrt{6}}{2} \right)}}{12} + \frac{\sqrt{6} \log{\left(\sin{\left(1 \right)} + \frac{\sqrt{6}}{2} \right)}}{12} - \frac{\sqrt{6} \left(\log{\left(- \sin{\left(1 \right)} + \frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12} + \frac{\sqrt{6} \left(\log{\left(\frac{\sqrt{6}}{2} \right)} + i \pi\right)}{12}$$

        /          /  ___         \\            /  ___\         /          /  ___\\            /  ___         \
    ___ |          |\/ 6          ||     ___    |\/ 6 |     ___ |          |\/ 6 ||     ___    |\/ 6          |
  \/ 6 *|pi*I + log|----- - sin(1)||   \/ 6 *log|-----|   \/ 6 *|pi*I + log|-----||   \/ 6 *log|----- + sin(1)|
        \          \  2           //            \  2  /         \          \  2  //            \  2           /
- ---------------------------------- - ---------------- + ------------------------- + -------------------------
                  12                          12                      12                          12

-sqrt(6)*(pi*i + log(sqrt(6)/2 - sin(1)))/12 - sqrt(6)*log(sqrt(6)/2)/12 + sqrt(6)*(pi*i + log(sqrt(6)/2))/12 + sqrt(6)*log(sqrt(6)/2 + sin(1))/12

Numerical answer [src]

0.343892904073394

0.343892904073394

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

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Identical expressions

Similar expressions

Integral of cosx/(3-2*sin^2x) dx

Limits of integration:

The graph:

Piecewise:

The solution