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Answer 1 · 2018-02-21T22:58:39

$k = 3$

We need to find the value for $k$ that satisfies the equation, so we will solve for $k$ .

However, first we need to integrate with respect to $x$ :

$int_0^k (4kx - 5k) dx = k^2$
$=> 1/2*4kx^2 - 5kx |_0^(x=k) = k^2$
$=> 2kx^2 - 5kx |_0^(x=k) = k^2$

Now substitute in the upper and lower bounds:

$=> 2kk^2 - 5kk - (2k*0^2 - 5k*0) = k^2$
$=> 2k^3 - 5k^2 = k^2$

Subtract $k^2$ from both sides:

$=> 2k^3 - 6k^2 = 0$

A solution is in sight, we just need to factor a bit:

$=> 2k^2*(k - 3) = 0$

We can see that for this entire thing to be zero, either $2k^2$ must be zero, or $k-3$ must be zero.

$2k^2 = 0$ implies $k=0$ so there is one solution for $k$ . However, since we want a non-zero answer:

$=>k - 3 = 0$
$=>k = 3$

Thus the answer is $k=3$ .