5.2-bo'lim : Noaniq integrallarni hisoblash
6. \( \displaystyle \int{{\sqrt[3]{w} + 10\,\,\sqrt[5]{{{w^3}}}\,dw}}\) ni baholang.
Maslahat: Ildizlarni kasr ko'rsatkichlariga aylantirishni unutmang.
Yechim
Biz birinchi navbatda ildizlarni kasr ko'rsatkichlariga aylantirishimiz kerak.
\[\int{{\sqrt[3]{w} + 10\,\,\sqrt[5]{{{w^3}}}\,dw}} = \int{{{w^{\,\frac{1}{3}}} + 10\,{{\left( {{w^3}} \right)}^{\frac{1}{5}}}\,dw}} = \int{{{w^{\,\frac{1}{3}}} + 10{w^{\,\frac{3}{5}}}\,dw}}\]2-qadam
Ildizlarni kasr ko'rsatkichlariga aylantirganimizdan so'ng, integralni baholashdan boshqa ishimiz qolmaydi.
\[\int{{\sqrt[3]{w} + 10\,\,\sqrt[5]{{{w^3}}}\,dw}} = \int{{{w^{\,\frac{1}{3}}} + 10{w^{\,\frac{3}{5}}}\,dw}} = \frac{3}{4}{w^{\,\frac{4}{3}}} + 10\left( {\frac{5}{8}} \right){w^{\,\frac{8}{5}}} + c = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{3}{4}{w^{\,\frac{4}{3}}} + \frac{{25}}{4}{w^{\,\frac{8}{5}}} + c}}\]“+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.