Are there multiple answers to integrals?
#int(1/(6x))dx#
Which integrates to:
#1/6ln(x) +c #
However, I also attempted to use another method of integrating and retrieved:
#1/6ln(6x)+c#
I am confused as they both derive the same-original equation yet are different answers. Is anyone able to help me understand why this is.
Thanks in advance.
Which integrates to:
However, I also attempted to use another method of integrating and retrieved:
I am confused as they both derive the same-original equation yet are different answers. Is anyone able to help me understand why this is.
Thanks in advance.
3 Answers
No, an integral will only give one solution (or in general, one class of solutions that differ by a constant term)
Explanation:
The first solution
I'm not sure how you arrived at the second answer, but there must have been a mistake with your computation.
Some times there are equivalent forms of solutions for integrals, but in this case the
Both of the the answer are consistent:
# 1/6ln(6x)+c = 1/6ln6 + 1/6lnx + c #
# \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 1/6lnx + A# where#A=c+1/6ln6#
Differentiation is a unique 1:1 function transformation, By FTC Integration is therefore unique but can differ by a constant (as the derivative of a constant is zero)
There are more than one answer in different form is
possible ,but not the different answer .
Here the different form means external look.!!
.But mathematically both the answers are same .
Explanation:
Before I answer see the both methods.
Let ,
So, your second answer is 100% correct.There is no mistake
with your computation.
There are more than one answer in different form is
possible ,but not the different answer .
Here the different form means external look.!!
.But mathematically both the answers are same .
Also, The subtraction of both the answers is always CONSTANT.
If we take,
Please see the illustrations carefully.