Cho hàm số f(x) có \[f\left( 2 \right) = 0\;\] và \[f\prime (x) = \frac{{x + 7}}{{\sqrt {2x - 3} }},\;\forall x \in (\frac{3}{2}; + \infty )\;\]. Biết rằng \[\mathop \smallint \limits_4^7 f\left( {\frac{x}{2}} \right)dx = \frac{a}{b}(a,b \in \mathbb{Z},b > 0,\frac{a}{b}\] là phân số tối giản). Khi đó a+b bằng:

Xét tích phân\[\mathop \smallint \limits_4^7 f\left( {\frac{x}{2}} \right)dx = \frac{a}{b}\]

Đặt\[t = \frac{x}{2} \Rightarrow dt = \frac{1}{2}dx \Leftrightarrow dx = 2dt\]  Đổi cận:\(\left\{ {\begin{array}{*{20}{c}}{x = 4 \Rightarrow t = 2}\\{x = 7 \Rightarrow t = \frac{7}{2}}\end{array}} \right.\)

Khi đó ta có:\[I = 2\mathop \smallint \limits_2^{\frac{7}{2}} f\left( t \right)dt\]

Đặt \(\left\{ {\begin{array}{*{20}{c}}{u = f(t)}\\{dv = dt}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}}{du = f\prime (t)dt}\\{v = t - \frac{7}{2}}\end{array}} \right.\) khi đó ta có:

\[I = 2\left( {\left( {t - \frac{7}{2}} \right)f(t)\mid _2^{\frac{\pi }{2}} - \int\limits_2^{\frac{7}{2}} {\left( {t - \frac{7}{2}} \right)} f\prime (t)dt} \right)\]

\(I = 2\left( {\frac{7}{2}f\left( 0 \right) - \int\limits_2^{\frac{7}{2}} {\left( {x - \frac{7}{2}} \right)f'\left( x \right)dx} } \right)\)

\(I = 2\left( {\frac{7}{2}f\left( 2 \right) - \int\limits_2^{\frac{7}{2}} {\left( {x - \frac{7}{2}} \right).\frac{{x + 7}}{{\sqrt {2x - 3} }}dx} } \right)\)

\(I = - 2\int\limits_2^{\frac{7}{2}} {\left( {x - \frac{7}{2}} \right)} .\frac{{x + 7}}{{\sqrt {2x - 3} }}dx\)

\(I = \frac{{236}}{{15}}\)

\[ \Rightarrow a = 236,b = 15\]

Vậy\[a + b = 236 + 15 = 251\]

Đáp án cần chọn là: B