Question

Find the value of'a'if int_(2)^(a)(x+1)dx=(7)/(2)​

Answer 1

Step-by-step explanation:

We have,

$\rm \int^{a}_{2}(x + 1)dx = \frac{7}{2}$

$\rm \implies \bigg [ \frac{ {x}^{2} }{2} + x \bigg] ^{a}_{2} = \frac{7}{2}$

$\rm \implies \frac{ {a}^{2} }{2} + a - \frac{ {2}^{2} }{2} - 2 = \frac{7}{2}$

$\rm \implies \frac{ {a}^{2} }{2} + a - \frac{ 4}{2} - 2 = \frac{7}{2}$

$\rm \implies \frac{ {a}^{2} }{2} + a - 2- 2 = \frac{7}{2}$

$\rm \implies \frac{ {a}^{2} }{2} + a - 4= \frac{7}{2}$

$\rm \implies \frac{ {a}^{2} }{2} + a = \frac{7}{2} + 4$

$\rm \implies \frac{ {a}^{2} }{2} + a = \frac{7 + 8}{2}$

$\rm \implies \frac{ {a}^{2} }{2} + a = \frac{15}{2}$

$\rm \implies {a}^{2} + 2a = 15$

$\rm \implies {a}^{2} + 2a - 15 = 0$

$\rm \implies {a}^{2} + 5a - 3a - 15 = 0$

$\rm \implies a (a + 5) - 3(a + 5 ) = 0$

$\rm \implies ( a - 3) (a + 5) = 0$

$\rm \implies a = 3 \: \: or \: \: a = - 5$

Since, upper limit must be greater than lower limit, so, $\rm\:a=3$