다음 적분을 구하라.
$$\int_0^8\sqrt[3]{x}dx$$
풀이
\(\displaystyle\int_0^8\sqrt[3]{x}dx=\int_0^8x^{\frac{1}{3}}dx=\left[\frac{3}{4}x^{\frac{4}{3}}\right]_0^8=\left[\frac{3}{4}x\sqrt[3]{x}\right]_0^8\)
\(\displaystyle=\frac{3}{4}(8)\sqrt[3]{8}-\frac{3}{4}(0)\sqrt[3]{0}=6\sqrt[3]{2^3}-0=6(2)=12\)
다음 적분을 구하라.
$$\int_1^2\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)dx$$
풀이
\(\displaystyle\int_1^2\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)dx=\int_1^2\left(x^{\frac{1}{2}}-x^{-\frac{1}{2}}\right)dx\)
\(\displaystyle=\left[\frac{2}{3}x^{\frac{3}{2}}-2x^{\frac{1}{2}}\right]_1^2=\left[\frac{2}{3}\sqrt{x^3}-2\sqrt{x}\right]_1^2\)
\(\displaystyle=\left[\frac{2}{3}x\sqrt{x}-2\sqrt{x}\right]_1^2=\frac{2}{3}(2)\sqrt{2}-2\sqrt{2}-\left(\frac{2}{3}(1)\sqrt{1}-2\sqrt{1}\right)\)
\(\displaystyle=\frac{4}{3}\sqrt{2}-2\sqrt{2}-\left(\frac{2}{3}-2\right)=-\frac{2}{3}\sqrt{2}-\left(-\frac{4}{3}\right)\)
\(\displaystyle=\frac{4}{3}-\frac{2}{3}\sqrt{2}\)
다음 적분을 구하라.
$$\int_0^1(1-x)^2dx$$
풀이
\(\displaystyle\int_0^1(1-x)^2dx=\left[-\frac{1}{3}(1-x)^3\right]_0^1\)
\(\displaystyle=-\frac{1}{3}(1-1)^3-\left(-\frac{1}{3}(1-0)^3\right)=\frac{1}{3}\)