What are all the values for k for which int_2^kx^5dx=0?
2 Answers
See below.
Explanation:
and
or
then finally
real values
complex values
k = +- 2
Explanation:
We require:
int_2^k x^5 \ dx = 0
Integrating we get:
[ x^6/6 ]_2^k = 0
:. 1/6 [ color(white)(""/"") x^6 ]_2^k = 0
:. 1/6(k^6-2^6) = 0
:. (k^3)^2-(2^3)^2 = 0
:. k^3 = +- 2^3
:. \ \k = +- 2 ,
Assuming that
Now, depending upon the context of the problem, one could argue that
Also, note that
Firstly, a property of definite integrals is that:
int_a^a f(x) = 0
so we can immediately establish
Secondly,
f(-x) = f(x)
and have rotational symmetry about the origin. as such, if
int_(a)^a \ f(x) = 0
so we can immediately establish
The integration and subsequent calculations do however prove that these are the only solutions!