Integral of sinx/sin3x dx

The solution

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

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

Integral(sin(x)/sin(3*x), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{\sin{\left(x \right)}}{\sin{\left(3 x \right)}} = \frac{\sin{\left(x \right)}}{- 4 \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}}$
Rewrite the integrand:

$\frac{\sin{\left(x \right)}}{- 4 \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}} = - \frac{1}{4 \sin^{2}{\left(x \right)} - 3}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{1}{4 \sin^{2}{\left(x \right)} - 3}\right)\, dx = - \int \frac{1}{4 \sin^{2}{\left(x \right)} - 3}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{3}}{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{3}}{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{3} \right)}}{6}$
So, the result is: $\frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{3}}{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{3}}{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{3} \right)}}{6}$
Now simplify:

$\frac{\sqrt{3} \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{3}}{3} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{3}}{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{3} \right)}\right)}{6}$
Add the constant of integration:

$\frac{\sqrt{3} \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{3}}{3} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{3}}{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{3} \right)}\right)}{6}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{3} \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{3}}{3} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + \frac{\sqrt{3}}{3} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + \sqrt{3} \right)}\right)}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$-{{\sqrt{3}\,\log \left({{4\,\sin ^2x+4\,\sqrt{3}\,\sin x+4\,\cos ^ 2x+4\,\cos x+4}\over{3}}\right)-\sqrt{3}\,\log \left({{4\,\sin ^2x+4 \,\sqrt{3}\,\sin x+4\,\cos ^2x-4\,\cos x+4}\over{3}}\right)-\sqrt{3} \,\log \left({{4\,\sin ^2x-4\,\sqrt{3}\,\sin x+4\,\cos ^2x+4\,\cos x +4}\over{3}}\right)+\sqrt{3}\,\log \left({{4\,\sin ^2x-4\,\sqrt{3}\, \sin x+4\,\cos ^2x-4\,\cos x+4}\over{3}}\right)}\over{12}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                    /          /              ___\\            /  ___\                                       /          /  ___\\                                                                      /  ___           \
                                ___ |          |            \/ 3 ||     ___    |\/ 3 |                                   ___ |          |\/ 3 ||                                                               ___    |\/ 3            |
    ___ /          /  ___\\   \/ 3 *|pi*I + log|-tan(1/2) + -----||   \/ 3 *log|-----|     ___    /  ___           \   \/ 3 *|pi*I + log|-----||     ___ /          /  ___           \\     ___    /  ___\   \/ 3 *log|----- + tan(1/2)|
  \/ 3 *\pi*I + log\\/ 3 //         \          \              3  //            \  3  /   \/ 3 *log\\/ 3  + tan(1/2)/         \          \  3  //   \/ 3 *\pi*I + log\\/ 3  - tan(1/2)//   \/ 3 *log\\/ 3 /            \  3             /
- ------------------------- - ------------------------------------- - ---------------- - --------------------------- + ------------------------- + ------------------------------------ + ---------------- + ---------------------------
              6                                 6                            6                        6                            6                                6                            6                        6

$$-{{\log \left({{4\,\sin ^21+4\,\sqrt{3}\,\sin 1+4\,\cos ^21+4\, \cos 1+4}\over{3}}\right)}\over{4\,\sqrt{3}}}+{{\log \left({{4\, \sin ^21+4\,\sqrt{3}\,\sin 1+4\,\cos ^21-4\,\cos 1+4}\over{3}} \right)}\over{4\,\sqrt{3}}}+{{\log \left({{4\,\sin ^21-4\,\sqrt{3}\, \sin 1+4\,\cos ^21+4\,\cos 1+4}\over{3}}\right)}\over{4\,\sqrt{3}}}- {{\log \left({{4\,\sin ^21-4\,\sqrt{3}\,\sin 1+4\,\cos ^21-4\,\cos 1 +4}\over{3}}\right)}\over{4\,\sqrt{3}}}$$

                                    /          /              ___\\            /  ___\                                       /          /  ___\\                                                                      /  ___           \
                                ___ |          |            \/ 3 ||     ___    |\/ 3 |                                   ___ |          |\/ 3 ||                                                               ___    |\/ 3            |
    ___ /          /  ___\\   \/ 3 *|pi*I + log|-tan(1/2) + -----||   \/ 3 *log|-----|     ___    /  ___           \   \/ 3 *|pi*I + log|-----||     ___ /          /  ___           \\     ___    /  ___\   \/ 3 *log|----- + tan(1/2)|
  \/ 3 *\pi*I + log\\/ 3 //         \          \              3  //            \  3  /   \/ 3 *log\\/ 3  + tan(1/2)/         \          \  3  //   \/ 3 *\pi*I + log\\/ 3  - tan(1/2)//   \/ 3 *log\\/ 3 /            \  3             /
- ------------------------- - ------------------------------------- - ---------------- - --------------------------- + ------------------------- + ------------------------------------ + ---------------- + ---------------------------
              6                                 6                            6                        6                            6                                6                            6                        6

$$- \frac{\sqrt{3} \log{\left(\tan{\left(\frac{1}{2} \right)} + \sqrt{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{3}}{3} \right)}}{6} - \frac{\sqrt{3} \log{\left(\frac{\sqrt{3}}{3} \right)}}{6} + \frac{\sqrt{3} \log{\left(\sqrt{3} \right)}}{6} - \frac{\sqrt{3} \left(\log{\left(\sqrt{3} \right)} + i \pi\right)}{6} - \frac{\sqrt{3} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{3}}{3} \right)} + i \pi\right)}{6} + \frac{\sqrt{3} \left(\log{\left(\frac{\sqrt{3}}{3} \right)} + i \pi\right)}{6} + \frac{\sqrt{3} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \sqrt{3} \right)} + i \pi\right)}{6}$$

Simplify

Numerical answer [src]

0.847473374471116

0.847473374471116

The graph

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

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Identical expressions

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Integral of sinx/sin3x dx

Limits of integration:

The graph:

Piecewise:

The solution