5.6-bo'lim : Aniq integralning ta'rifi
5. \( \displaystyle \int_{6}^{{11}}{{f\left( x \right)\,dx}} = - 7\) va \( \displaystyle \int_{6}^{{11}}{{g\left( x \right)\,dx}} = 24\) berilgan \( \displaystyle \int_{6}^{{11}}{{6g\left( x \right) - 10f\left( x \right)\,dx}}\) qiymatini aniqlang.
Yechim
Haqiqatan ham bu muammoni ishlatishdan boshqa ko'p narsa yo'q xususiyatlari ushbu bo'limning eslatmalaridan biz berilgan intervallarni olinmaguncha, biz berilgan qiymatlardan foydalanamiz.
\[\begin{align*}\int_{6}^{{11}}{{6g\left( x \right) - 10f\left( x \right)\,dx}} & = \int_{6}^{{11}}{{6g\left( x \right)\,dx}} - \int_{6}^{{11}}{{10f\left( x \right)\,dx}} & \hspace{0.5in} & {\mbox{Property 4}}\\ & = 6\int_{6}^{{11}}{{g\left( x \right)\,dx}} - 10\int_{6}^{{11}}{{f\left( x \right)\,dx}} & \hspace{0.5in} & {\mbox{Property 3}}\\ & = 6\left( {24} \right) - 10\left( { - 7} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{214}} & & \end{align*}\]