Question

If the value of \[\int_1^k {(2x - 3)dx = 12} \], then find the value of k.
\[({\text{a}})\] \[ - 2\] and\[{\text{5}}\]
\[{\text{(b)}}\] \[{\text{5}}\]and \[{\text{2}}\]
\[{\text{(c)}}\] \[{\text{2}}\] and\[{\text{ - 5}}\]
\[{\text{(d)}}\]None of these

Answer 1

Hint: Evaluate the given integral carefully without missing any term in between.

We have the given integral as
\[\int_1^k {(2k - 3)dx} \]
After integrating the above equation, we get,
\[ = [{x^2} - 3x]_1^k\]
\[ = ({k^2} - 3k) - (1 - 3)\]
\[ = {k^2} - 3k + 2\]
According to the question,
We are given that the value of the given integral is equal to $12$,
Therefore, we get
\[{k^2} - 3k + 2 = 12\]
\[{k^2} - 3k - 10 = 0\]
This equation can be re written in the form as
\[{k^2} - 5k + 2k - 10 = 0\]
\[k(k - 5) + 2(k - 5) = 0\]
\[(k + 2)(k - 5) = 0\]
\[\therefore k = - 2,5\]
Therefore, the required solution is \[({\text{a}})\] \[ - 2\] and\[{\text{5}}\].

Note: In these types of questions, the given integral is solved, then equated to the values given in the question, which on evaluation gives the value of the required variable.