QJEEM18Ma0f85: JEE Main 2018 Mathematics - Class 12 JEEM18Ma0f85 Solution: If    (where C is a constant of integration), then…

Question

Verified Solution • 2025–26 Session moderate

If

\int {{{2x + 5} \over {\sqrt {7 - 6x - {x^2}} }}} \,\,dx = A\sqrt {7 - 6x - {x^2}} + B{\sin ^{ - 1}}\left( {{{x + 3} \over 4}} \right) + C

(where C is a constant of integration), then the ordered pair (A, B) is equal to :

Accepted Answer

We can write,

7 - 6x - x² = 16 - (x + 3)²

and

{d \over {dx}}\left( {7 - 6x - {x^2}} \right) = - (2x + 6)

So,

\int {{{2x + 5} \over {\sqrt {7 - 6x - {x^2}} }}} dx

=

\int {{{2x + 6 - 1} \over {\sqrt {7 - 6x - {x^2}} }}} dx

=

\int {{{2x + 6} \over {\sqrt {7 - 6x - {x^2}} }}} dx

-

\int {{1 \over {\sqrt {16 - {{(x + 3)}^2}} }}} dx

=

\int {{{2x + 6} \over {\sqrt {7 - 6x - {x^2}} }}} dx

-

\int {{1 \over {\sqrt {{{\left( 4 \right)}^2} - {{(x + 3)}^2}} }}} dx

=

- 2\sqrt {7 - 6x - {x^2}}

-

{\sin ^{ - 1}}\left( {{{x + 3} \over 4}} \right)

+ C

By comparing with the given equation in the question we get,

A = - 2 and B = - 1

QJEEM18Ma0f85: JEE Main 2018 Mathematics - Class 12 JEEM18Ma0f85 Solution: If &nbsp;&nbsp; (where C is a constant of integration), then…