적분 – 한철수

대학수학 연습문제

다음 적분을 구하라.

$$\int_{-1}^2\vert x-1\vert dx$$

풀이

$\displaystyle\int_{-1}^2\vert x-1\vert dx=\int_1^2(x-1)dx+\int_{-1}^1(-x+1)dx$

$\displaystyle=\left[\frac{1}{2}x^2-x\right]_1^2+\left[-\frac{1}{2}x^2+x\right]_{-1}^1$

$\displaystyle=\frac{2^2}{2}-2-\left(\frac{1^2}{2}-1\right)-\frac{1^2}{2}+1-\left(-\frac{(-1)^2}{2}-1\right)$

$\displaystyle=0-\left(-\frac{1}{2}\right)+\frac{1}{2}-\left(-\frac{3}{2}\right)=\frac{5}{2}$

다음 적분을 구하라.

$$\int_1^3\frac{u^4-8}{u^2}du$$

풀이

$\displaystyle\int_1^3\frac{u^4-8}{u^2}du=\int_1^3\left(u^2-8u^{-2}\right)du$

$\displaystyle=\left[\frac{1}{3}u^3-8(-1)u^{-1}\right]_1^3=\left[\frac{u^3}{3}+\frac{8}{u}\right]_1^3$

$\displaystyle=\frac{3^3}{3}+\frac{8}{3}-\left(\frac{1^3}{3}+\frac{8}{1}\right)=\frac{35}{3}-\frac{25}{3}=\frac{10}{3}$

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

$\displaystyle\int_0^1(1-x)^2dx=\left[-\frac{1}{3}(1-x)^3\right]_0^1$

$\displaystyle=-\frac{1}{3}(1-1)^3-\left(-\frac{1}{3}(1-0)^3\right)=\frac{1}{3}$