@Carl Love 

I apologize for not expressing my question clearly. Your solution is exactly what I want to know!

@Carl Love 

plot3dlines.mw

Thank you so much!

Another problem I encounter is how to plot lines instead of a surface in plot3d when i_t is integer and θ_t is in a range from -Pi to Pi? 

@Markiyan Hirnyk 

You mean whether getting stuck depends on the operating system? I ran the procedure in a Win 7 computer workstation, and  staying in "Evaluating..." still happened if I didn't write the integral in the form C.

@Carl Love I find the option! It's very kind of you!

@Markiyan Hirnyk 

Thank you for your help! I tried the way you put forward and it is helpful!

@Carl Love 

I want to see the results just for checking each step of the procedure.

You are right! I should find another way to estimate the results.

By the way, could you please teach me how to display the result whose length exceeds limit of 1000000? I am curious about the way to display：）

@Carl Love 

Unknown_Variables_in_Definite_Integral.mw

There are 3 unknown variables in the definite integral, so I use indefinite integral function and eval to get the result with no integral symbol inside.

stuck_in_integral.mw

@I_Mariusz 

When I write integral expressions in the form A displayed in the file I uploaded,evaluating always gets stuck in calculating the integral.
When in the form B, it sometimes gets stuck. But when in the form C, it hardly gets stuck. I feel a little confused.

@Carl Love Thank you for your advice!

@I_Mariusz 

Is it possible to get the approximation which is an expression containing unknown variables i_t, h₀, theta[alpha], alpha[c]?

@I_Mariusz Thank you so much!

@Kitonum @vv 

I'm sorry I removed several lines of codes and kept only 4 lines left. I feel sorry too that I didn't express my question clearly. p[t] is a function depending on theta  and  r. I have defined it before integrating. r[5], r[6]  and  theta[tt]  are constants I have defined too.

When calculating the integral, I found the running time of the second method is much shorter than that of the first method. I am worried whether short running time means lower precision in this situation.

@acer I really appreciate your help! I find the line by following your steps!

@Markiyan Hirnyk @Adri van der Meer 

Though I don't know the reason why the Tutor popped up, I have solved this problem by rewriting the codes which are the trigger. Thank both of you for your patience!

About_the_TUTOR_POP.mw

I deleted some codes step by step, then I find the left codes which I uploaded may have something wrong.