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Solving integrals/ sin^2x/(sinx+2cosx)

Integral of sin^2x/(sinx+2cosx) dx

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  1                     
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 |       sin (x)        
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 |  sin(x) + 2*cos(x)   
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∫01sin⁡2(x)sin⁡(x)+2cos⁡(x) dx\int\limits_{0}^{1} \frac{\sin^{2}{\left(x \right)}}{\sin{\left(x \right)} + 2 \cos{\left(x \right)}}\, dx 
Integral(sin(x)^2/(sin(x) + 2*cos(x)), (x, 0, 1))
The answer (Indefinite) [src] 
  /                                                                          /        ___         \              /        ___         \                      /        ___         \                      /        ___         \
 |                                                       /x\          ___    |  1   \/ 5       /x\|       ___    |  1   \/ 5       /x\|       ___    2/x\    |  1   \/ 5       /x\|       ___    2/x\    |  1   \/ 5       /x\|
 |         2                                       20*tan|-|      4*\/ 5 *log|- - - ----- + tan|-||   4*\/ 5 *log|- - + ----- + tan|-||   4*\/ 5 *tan |-|*log|- - - ----- + tan|-||   4*\/ 5 *tan |-|*log|- - + ----- + tan|-||
 |      sin (x)                      10                  \2/                 \  2     2        \2//              \  2     2        \2//               \2/    \  2     2        \2//               \2/    \  2     2        \2//
 | ----------------- dx = C - --------------- - --------------- - --------------------------------- + --------------------------------- - ----------------------------------------- + -----------------------------------------
 | sin(x) + 2*cos(x)                     2/x\              2/x\                       2/x\                                2/x\                                    2/x\                                        2/x\             
 |                            25 + 25*tan |-|   25 + 25*tan |-|            25 + 25*tan |-|                     25 + 25*tan |-|                         25 + 25*tan |-|                             25 + 25*tan |-|             
/                                         \2/               \2/                        \2/                                 \2/                                     \2/                                         \2/
∫sin⁡2(x)sin⁡(x)+2cos⁡(x) dx=C+45log⁡(tan⁡(x2)−12+52)tan⁡2(x2)25tan⁡2(x2)+25+45log⁡(tan⁡(x2)−12+52)25tan⁡2(x2)+25−45log⁡(tan⁡(x2)−52−12)tan⁡2(x2)25tan⁡2(x2)+25−45log⁡(tan⁡(x2)−52−12)25tan⁡2(x2)+25−20tan⁡(x2)25tan⁡2(x2)+25−1025tan⁡2(x2)+25\int \frac{\sin^{2}{\left(x \right)}}{\sin{\left(x \right)} + 2 \cos{\left(x \right)}}\, dx = C + \frac{4 \sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25} + \frac{4 \sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25} - \frac{4 \sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{5}}{2} - \frac{1}{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25} - \frac{4 \sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \frac{\sqrt{5}}{2} - \frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25} - \frac{20 \tan{\left(\frac{x}{2} \right)}}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25} - \frac{10}{25 \tan^{2}{\left(\frac{x}{2} \right)} + 25}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.00.5
Plot the graph f(x)
The answer [src] 
                                                       /        ___\           /          /      ___\\           /          /      ___           \\              /        ___           \                     /          /      ___           \\                        /        ___           \
                                                ___    |  1   \/ 5 |       ___ |          |1   \/ 5 ||       ___ |          |1   \/ 5            ||       ___    |  1   \/ 5            |       ___    2      |          |1   \/ 5            ||       ___    2         |  1   \/ 5            |
                                            4*\/ 5 *log|- - + -----|   4*\/ 5 *|pi*I + log|- + -----||   4*\/ 5 *|pi*I + log|- + ----- - tan(1/2)||   4*\/ 5 *log|- - + ----- + tan(1/2)|   4*\/ 5 *tan (1/2)*|pi*I + log|- + ----- - tan(1/2)||   4*\/ 5 *tan (1/2)*log|- - + ----- + tan(1/2)|
2           10             20*tan(1/2)                 \  2     2  /           \          \2     2  //           \          \2     2             //              \  2     2             /                     \          \2     2             //                        \  2     2             /
- - ----------------- - ----------------- - ------------------------ + ------------------------------- - ------------------------------------------ + ----------------------------------- - ---------------------------------------------------- + ---------------------------------------------
5              2                   2                   25                             25                                        2                                         2                                             2                                                   2                   
    25 + 25*tan (1/2)   25 + 25*tan (1/2)                                                                            25 + 25*tan (1/2)                         25 + 25*tan (1/2)                             25 + 25*tan (1/2)                                   25 + 25*tan (1/2)
−20tan⁡(12)25tan⁡2(12)+25−1025tan⁡2(12)+25+45log⁡(−12+tan⁡(12)+52)tan⁡2(12)25tan⁡2(12)+25+45log⁡(−12+tan⁡(12)+52)25tan⁡2(12)+25−45log⁡(−12+52)25+25−45(log⁡(−tan⁡(12)+12+52)+iπ)25tan⁡2(12)+25−45(log⁡(−tan⁡(12)+12+52)+iπ)tan⁡2(12)25tan⁡2(12)+25+45(log⁡(12+52)+iπ)25- \frac{20 \tan{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{10}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{5}}{2} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{5}}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{25} + \frac{2}{5} - \frac{4 \sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{4 \sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{25} 
=
= 
                                                       /        ___\           /          /      ___\\           /          /      ___           \\              /        ___           \                     /          /      ___           \\                        /        ___           \
                                                ___    |  1   \/ 5 |       ___ |          |1   \/ 5 ||       ___ |          |1   \/ 5            ||       ___    |  1   \/ 5            |       ___    2      |          |1   \/ 5            ||       ___    2         |  1   \/ 5            |
                                            4*\/ 5 *log|- - + -----|   4*\/ 5 *|pi*I + log|- + -----||   4*\/ 5 *|pi*I + log|- + ----- - tan(1/2)||   4*\/ 5 *log|- - + ----- + tan(1/2)|   4*\/ 5 *tan (1/2)*|pi*I + log|- + ----- - tan(1/2)||   4*\/ 5 *tan (1/2)*log|- - + ----- + tan(1/2)|
2           10             20*tan(1/2)                 \  2     2  /           \          \2     2  //           \          \2     2             //              \  2     2             /                     \          \2     2             //                        \  2     2             /
- - ----------------- - ----------------- - ------------------------ + ------------------------------- - ------------------------------------------ + ----------------------------------- - ---------------------------------------------------- + ---------------------------------------------
5              2                   2                   25                             25                                        2                                         2                                             2                                                   2                   
    25 + 25*tan (1/2)   25 + 25*tan (1/2)                                                                            25 + 25*tan (1/2)                         25 + 25*tan (1/2)                             25 + 25*tan (1/2)                                   25 + 25*tan (1/2)
−20tan⁡(12)25tan⁡2(12)+25−1025tan⁡2(12)+25+45log⁡(−12+tan⁡(12)+52)tan⁡2(12)25tan⁡2(12)+25+45log⁡(−12+tan⁡(12)+52)25tan⁡2(12)+25−45log⁡(−12+52)25+25−45(log⁡(−tan⁡(12)+12+52)+iπ)25tan⁡2(12)+25−45(log⁡(−tan⁡(12)+12+52)+iπ)tan⁡2(12)25tan⁡2(12)+25+45(log⁡(12+52)+iπ)25- \frac{20 \tan{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{10}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{5}}{2} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \tan{\left(\frac{1}{2} \right)} + \frac{\sqrt{5}}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{4 \sqrt{5} \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{25} + \frac{2}{5} - \frac{4 \sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} - \frac{4 \sqrt{5} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{25 \tan^{2}{\left(\frac{1}{2} \right)} + 25} + \frac{4 \sqrt{5} \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{25} 
2/5 - 10/(25 + 25*tan(1/2)^2) - 20*tan(1/2)/(25 + 25*tan(1/2)^2) - 4*sqrt(5)*log(-1/2 + sqrt(5)/2)/25 + 4*sqrt(5)*(pi*i + log(1/2 + sqrt(5)/2))/25 - 4*sqrt(5)*(pi*i + log(1/2 + sqrt(5)/2 - tan(1/2)))/(25 + 25*tan(1/2)^2) + 4*sqrt(5)*log(-1/2 + sqrt(5)/2 + tan(1/2))/(25 + 25*tan(1/2)^2) - 4*sqrt(5)*tan(1/2)^2*(pi*i + log(1/2 + sqrt(5)/2 - tan(1/2)))/(25 + 25*tan(1/2)^2) + 4*sqrt(5)*tan(1/2)^2*log(-1/2 + sqrt(5)/2 + tan(1/2))/(25 + 25*tan(1/2)^2)
Numerical answer [src] 
0.129328850515895
0.129328850515895

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

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