If _ x 0 e¹² - (1+ax)^ b x x = 18 e¹² , where a and b are positive constants, then the value of a+b is equal to

Step-by-step solution

Let L = _ x 0 e¹² - (1+ax)^ b x x . Rewrite the term (1+ax)^ b x as e^ b x (1+ax) . Using the Maclaurin series expansion for (1+ax) : (1+ax) = ax - (ax)^2 2 + = ax - a^2 x^2 2 + Multiply by b x : b x (1+ax) = ab - a^2 b x 2 + Thus, the exponential term becomes: e^ ab - a^2 b x 2 + = e^ ab e^ - a^2 b x 2 + = e^ ab (1 - a^2 b x 2 + ) Substitute this into the limit expression: L = _ x 0 e¹² - e^ ab (1 - a^2 b x 2 + ) x For the limit to exist and be finite, the constant term in the numerator must be zero. Therefore, e¹