Something Interesting I Realized About Integrals

[okkvlt]

suppose i want to find the following integral:
7
$$\int$$x dx
3

now suppose for some demented reason i decided not to do it straightforward and get (49-9)/2=20

instead i use the substitution x=u²+4u+5
giving
u₁
$$\int$$(u²+4u+5)(2u+4)du
u₀
u₁
$$\int$$2u³+12u²+26u+20 du
u₀
the indefinite integral is .5u⁴+4u³+13u²+20u

now its time to find the limits of integration,
by the quadratic formula,
u₁=(-4$$\pm$$$$\sqrt{24}$$)/2=0.44948974278318,-4.44948974278318

u₀=(-4$$\pm$$$$\sqrt{8}$$)/2=-0.5857864376269, -3.4142135623731
what happens when you enter these limits of integration?
Here is what's fascinating:
if i use both values of u₁ as the limits of integration, i get 0 (the case is the same with using both values of u₀).
if i use either value of u₁ as the upper limit and either value of u₀ as the lower limit, i get the correct answer of 20. there is a total of 4 ways to correctly get the answer. it doesn't matter which of the values i use as long as i use a value of u1 as the upper limit and a value of u0 as the lower limit.

you can go ahead and test it to see what i mean.

 

[HallsofIvy]

suppose i want to find the following integral:
7
$$\int$$x dx
3

now suppose for some demented reason i decided not to do it straightforward and get (49-9)/2=20

instead i use the substitution x=u²+4u+5
giving
u₁
$$\int$$(u²+4u+5)(2u+4)du
u₀
u₁
$$\int$$2u³+12u²+26u+20 du
u₀
the indefinite integral is .5u⁴+4u³+13u²+20u

now its time to find the limits of integration,
by the quadratic formula,
u₁=(-4$$\pm$$$$\sqrt{24}$$)/2=0.44948974278318,-4.44948974278318

u₀=(-4$$\pm$$$$\sqrt{8}$$)/2=-0.5857864376269, -3.4142135623731



what happens when you enter these limits of integration?
Here is what's fascinating:
if i use both values of u₁ as the limits of integration, i get 0 (the case is the same with using both values of u₀).

Yes, of course. The two values of u₁ both correspond to x= 7 and the integral of any function from "7" to "7" is 0.

if i use either value of u₁ as the upper limit and either value of u₀ as the lower limit, i get the correct answer of 20. there is a total of 4 ways to correctly get the answer. it doesn't matter which of the values i use as long as i use a value of u1 as the upper limit and a value of u0 as the lower limit.

Somewhere in your calculation you are doing the equivalent putting your values of u₀ and u₁ into the formula u²+ 4u+ 5 and getting values of 7 for both values of u₁ and 3 for both values of u₀. This behavior is eactly what you should expect.

you can go ahead and test it to see what i mean.