suppose i want to find the following integral:(adsbygoogle = window.adsbygoogle || []).push({});

7

[tex]\int[/tex]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=u^{2}+4u+5

giving

u_{1}

[tex]\int[/tex](u^{2}+4u+5)(2u+4)du

u_{0}

u_{1}

[tex]\int[/tex]2u^{3}+12u^{2}+26u+20 du

u_{0}

the indefinite integral is .5u^{4}+4u^{3}+13u^{2}+20u

now its time to find the limits of integration,

by the quadratic formula,

u_{1}=(-4[tex]\pm[/tex][tex]\sqrt{24}[/tex])/2=0.44948974278318,-4.44948974278318

u_{0}=(-4[tex]\pm[/tex][tex]\sqrt{8}[/tex])/2=-0.5857864376269, -3.4142135623731

what happens when you enter these limits of integration?

Here is whats fascinating:

if i use both values of u_{1}as the limits of integration, i get 0 (the case is the same with using both values of u_{0}).

if i use either value of u_{1}as the upper limit and either value of u_{0}as the lower limit, i get the correct answer of 20. there is a total of 4 ways to correctly get the answer. it doesnt 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.

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# Something interesting i realized. (more than one set of limits, change of variables)

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