تكامل التابع الجذري

تكامل التابع الجذري
( T: Köklü Fonksiyonun İntegralı )
(E: Integral of Radical Function)

\[\int \sqrt[n]{x}~dx=\int x^{\frac{1}{n}}~dx=\frac{1}{\frac{1}{n}+1}~x^{\frac{1}{n}+1}+c\]
\[=\frac{n}{n+1}~x^{\frac{n+1}{n}}+c~~, ~,~c \in \Bbb R\]

مثال1:
أوجد تكامل التابع:
\[f(x)=\sqrt[3]{x}\]
الحل:
\[\int \sqrt[3]{x}~dx=\int x^{\frac{1}{3}}~dx=\frac{1}{\frac{1}{3}+1}~x^{\frac{1}{3}+1}+c\]
\[=\frac{3}{4}~x^{\frac{4}{3}}+c~~, ~,~c \in \Bbb R\]
مثال 2:
أوجد تكامل التابع:
\[f(x)=5\sqrt[4]{x}+\sqrt[7]{x}\]
الحل:
\[\int \left(5\sqrt[4]{x}+\sqrt[7]{x}\right)~dx=\]
بتطبيق خواص التكامل نجد أن:
\[=5 \cdot \int \sqrt[4]{x}~dx+\int \sqrt[7]{x}~dx\]
\[=5 \cdot \int x^{\frac{1}{4}}~dx+\int x^{\frac{1}{7}}~dx\]
\[=5 \cdot \left(\frac{1}{\frac{1}{4}+1}~x^{\frac{1}{4}+1}\right)+\left(\frac{1}{\frac{1}{7}+1}~x^{\frac{1}{7}+1}\right)+c\]
\[=5 \cdot \frac{4}{5}~x^{\frac{5}{4}}+\frac{7}{8}~x^{\frac{8}{7}}+c\]
\[=4~x^{\frac{5}{4}}+\frac{7}{8}~x^{\frac{8}{7}}+c~~, ~,~c \in \Bbb R\]

\[\int \acute f(x)\cdot \sqrt[n]{f(x)}~dx=\int \acute f(x)\cdot {f(x)}^\frac{1}{n}~dx\]
\[=\frac{1}{\frac{1}{n}+1}~{f(x)}^{\frac{1}{n}+1}+c=\frac{n}{n+1}~{f(x)}^{\frac{n+1}{n}}+c~~, ~,~c \in \Bbb R\]

مثال 3:
أوجد تكامل التابع:
\[f(x)=2x\sqrt[3]{x^2}\]
الحل:
\[\int 2x\sqrt[3]{x^2}~dx=\int (2x)\cdot {(x^2)}^\frac{1}{3}~dx\]
\[=\frac{1}{\frac{1}{3}+1}~{(x^2)}^{\frac{1}{3}+1}+c=\frac{3}{4}~{(x^2)}^{\frac{4}{3}}+c~~, ~,~c \in \Bbb R\]
حيث
\[(x^2)’=2x\]