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  • (a*sin^ three (x/ three))^ two
  • (a multiply by sinus of cubed (x divide by 3)) squared
  • (a multiply by sinus of to the power of three (x divide by three)) to the power of two
  • (a*sin3(x/3))2
  • a*sin3x/32
  • (a*sin³(x/3))²
  • (a*sin to the power of 3(x/3)) to the power of 2
  • (asin^3(x/3))^2
  • (asin3(x/3))2
  • asin3x/32
  • asin^3x/3^2
  • (a*sin^3(x divide by 3))^2
  • (a*sin^3(x/3))^2dx
Solving integrals/ (a*sin^3(x/3))^2

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

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The solution

You have entered [src] 
 pi                
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 |             2   
 |  /     3/x\\    
 |  |a*sin |-||  dx
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pi                 
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2
∫π2π(asin⁡3(x3))2 dx\int\limits_{\frac{\pi}{2}}^{\pi} \left(a \sin^{3}{\left(\frac{x}{3} \right)}\right)^{2}\, dx 
Integral((a*sin(x/3)^3)^2, (x, pi/2, pi))
The answer (Indefinite) [src] 
  /                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           
 |                                                                2    5/x\                                                                               2    3/x\                                                                               2    /x\                                                                                                                                                                       2    11/x\                                                                               2    9/x\                                                                               2    7/x\                                                                              2    12/x\                                                                               2    2/x\                                                                               2    10/x\                                                                              2    4/x\                                                                               2    8/x\                                                                                2    6/x\                                  
 |            2                                              396*a *tan |-|                                                                          170*a *tan |-|                                                                           30*a *tan|-|                                                                                    2                                                                              30*a *tan  |-|                                                                          170*a *tan |-|                                                                          396*a *tan |-|                                                                         5*x*a *tan  |-|                                                                         30*x*a *tan |-|                                                                         30*x*a *tan  |-|                                                                        75*x*a *tan |-|                                                                         75*x*a *tan |-|                                                                         100*x*a *tan |-|                                  
 | /     3/x\\                                                          \6/                                                                                     \6/                                                                                    \6/                                                                               5*x*a                                                                                          \6/                                                                                     \6/                                                                                     \6/                                                                                     \6/                                                                                     \6/                                                                                      \6/                                                                                    \6/                                                                                     \6/                                                                                      \6/                                  
 | |a*sin |-||  dx = C - ------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------
 | \      \3//                      12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\              12/x\         2/x\         10/x\          4/x\          8/x\          6/x\
 |                       16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|   16 + 16*tan  |-| + 96*tan |-| + 96*tan  |-| + 240*tan |-| + 240*tan |-| + 320*tan |-|
/                                     \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/                \6/          \6/           \6/           \6/           \6/           \6/
∫(asin⁡3(x3))2 dx=C+5a2xtan⁡12(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16+30a2xtan⁡10(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16+75a2xtan⁡8(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16+100a2xtan⁡6(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16+75a2xtan⁡4(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16+30a2xtan⁡2(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16+5a2x16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16+30a2tan⁡11(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16+170a2tan⁡9(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16+396a2tan⁡7(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16−396a2tan⁡5(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16−170a2tan⁡3(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16−30a2tan⁡(x6)16tan⁡12(x6)+96tan⁡10(x6)+240tan⁡8(x6)+320tan⁡6(x6)+240tan⁡4(x6)+96tan⁡2(x6)+16\int \left(a \sin^{3}{\left(\frac{x}{3} \right)}\right)^{2}\, dx = C + \frac{5 a^{2} x \tan^{12}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{30 a^{2} x \tan^{10}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{75 a^{2} x \tan^{8}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{100 a^{2} x \tan^{6}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{75 a^{2} x \tan^{4}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{30 a^{2} x \tan^{2}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{5 a^{2} x}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{30 a^{2} \tan^{11}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{170 a^{2} \tan^{9}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} + \frac{396 a^{2} \tan^{7}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} - \frac{396 a^{2} \tan^{5}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} - \frac{170 a^{2} \tan^{3}{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16} - \frac{30 a^{2} \tan{\left(\frac{x}{6} \right)}}{16 \tan^{12}{\left(\frac{x}{6} \right)} + 96 \tan^{10}{\left(\frac{x}{6} \right)} + 240 \tan^{8}{\left(\frac{x}{6} \right)} + 320 \tan^{6}{\left(\frac{x}{6} \right)} + 240 \tan^{4}{\left(\frac{x}{6} \right)} + 96 \tan^{2}{\left(\frac{x}{6} \right)} + 16}
Simplify
The answer [src] 
   /       ___       \      /      ___       \
 2 |  27*\/ 3    5*pi|    2 |  9*\/ 3    5*pi|
a *|- -------- + ----| - a *|- ------- + ----|
   \     64       16 /      \     32      32 /
−a2(−9332+5π32)+a2(−27364+5π16)- a^{2} \left(- \frac{9 \sqrt{3}}{32} + \frac{5 \pi}{32}\right) + a^{2} \left(- \frac{27 \sqrt{3}}{64} + \frac{5 \pi}{16}\right) 
=
= 
   /       ___       \      /      ___       \
 2 |  27*\/ 3    5*pi|    2 |  9*\/ 3    5*pi|
a *|- -------- + ----| - a *|- ------- + ----|
   \     64       16 /      \     32      32 /
−a2(−9332+5π32)+a2(−27364+5π16)- a^{2} \left(- \frac{9 \sqrt{3}}{32} + \frac{5 \pi}{32}\right) + a^{2} \left(- \frac{27 \sqrt{3}}{64} + \frac{5 \pi}{16}\right) 
a^2*(-27*sqrt(3)/64 + 5*pi/16) - a^2*(-9*sqrt(3)/32 + 5*pi/32)

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

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