5.2-bo'lim : Noaniq integrallarni hisoblash
7. \( \displaystyle \int{{\sqrt {{x^7}} - 7\,\sqrt[6]{{{x^5}}} + 17\,\,\sqrt[3]{{{x^{10}}}}\,dx}}\) ni baholang.
Maslahat: Ildizlarni kasr ko'rsatkichlariga aylantirishni unutmang.
Yechim
Biz birinchi navbatda ildizlarni kasr ko'rsatkichlariga aylantirishimiz kerak.
\[\int{{\sqrt {{x^7}} - 7\,\sqrt[6]{{{x^5}}} + 17\,\,\sqrt[3]{{{x^{10}}}}\,dx}} = \int{{{x^{\frac{7}{2}}} - 7{{\left( {{x^5}} \right)}^{\frac{1}{6}}} + 17{{\left( {{x^{10}}} \right)}^{\frac{1}{3}}}\,dx}} = \int{{{x^{\frac{7}{2}}} - 7{x^{\frac{5}{6}}} + 17{x^{\frac{{10}}{3}}}\,dx}}\]2-qadam
Ildizlarni kasr ko'rsatkichlariga aylantirganimizdan so'ng, integralni baholashdan boshqa ishimiz qolmaydi.
\[\begin{align*}\int{{\sqrt {{x^7}} - 7\,\sqrt[6]{{{x^5}}} + 17\,\,\sqrt[3]{{{x^{10}}}}\,dx}} & = \int{{{x^{\frac{7}{2}}} - 7{x^{\frac{5}{6}}} + 17{x^{\frac{{10}}{3}}}\,dx}}\\ & = \frac{2}{9}{x^{\frac{9}{2}}} - 7\left( {\frac{6}{{11}}} \right){x^{\frac{{11}}{6}}} + 17\left( {\frac{3}{{13}}} \right){x^{\frac{{13}}{3}}} + c = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{2}{9}{x^{\frac{9}{2}}} - \frac{{42}}{{11}}{x^{\frac{{11}}{6}}} + \frac{{51}}{{13}}{x^{\frac{{13}}{3}}} + c}}\end{align*}\]“+c” ni qo‘shishni unutmang, chunki biz integralni olish uchun qaysi funktsiyani farqlaganimizni so‘raayotganimizni bilamiz va doimiyning hosilasi nolga teng va shuning uchun javobga buni qo‘shishimiz kerak.