My question is as follows: In learning integration, I was taught to change the limits of integration when using u-substitution and solve accordingly. However, now that I am solving definite integrals involving inverse trigonometric functions, my calculus book is no longer making substitutions for the limits of integration based on u-substitutions that I am making? Please explain..........
But my problem is that when the book was teaching u-substitution, I understood that unless limits of integration were changed that the final answer would have to be changed back in terms of x. However, in the chapter on integration of inverse trig functions, the book uses u-substitution but doesn't change the limits of integration in terms of u and just integrates as if the integral was written in terms of x. So how is it that in some instances, u-substitution warrants changes in limits of integration, and in other instances, u-substitution is used but the limits of integration are not changed to compensate for the u-substitution?
Definite Integral evaluated from 0 to 1/6: 1/(1-9x^2)^1/2 dx requires u-substitution of u=3x, but the final answer does not reflect a change in limits of integration from 0 to 1/2 based on the u-substitution of u=3x. The integral is evaluated from 0 to 1/6 even though u-substitution was employed and the answer is pi/18. So again, my question is this: in other instances, the limits of integration would be changed to (0 to 3(1/6) or (0 to 1/2) based on the u-substitution, but in this case they are not changed and the integration is done based on the original limits of integration even thought u-substitution was used?????
If you are saying they integrated by making the substitution u= 3x so that du= 3 dx and then writing it as
which is about 0.05582, then that is just wrong.
Correct is either
which is about 0.17453
or replacing [tex]sin^{-1}(u)[tex] with to get and again, about 0.17453.