Nếu \(\int\limits_{1}^{2}{f\left( x \right)dx=3}\) và \(\int\limits_{6}^{12}{f\left( \frac{x}{3} \right)dx}=2\) thì \(\int\limits_{1}^{4}{f\left( x \right)dx}\) bằng

* Hướng dẫn giải

Ta có

\(\int\limits_6^{12} {f\left( {\frac{x}{3}} \right)dx}  = 3\int\limits_6^{12} {f\left( {\frac{x}{3}} \right)d\left( {\frac{x}{3}} \right) = 3} \int\limits_2^4 {f(t)dt}  = 3\int\limits_2^4 {f(x)dx} \)

Suy ra \(\int\limits_2^4 {f\left( x \right)dx}  = \frac{2}{3}\)

Vậy \(\int\limits_1^4 {f(x)dx}  = \int\limits_1^2 {f(x)dx}  + \int\limits_2^4 {f(x)dx}  = 3 + \frac{2}{3} = \frac{{11}}{3}\)