\(\displaystyle f(x)=\frac{1}{4}\sqrt{x}-\frac{1}{x^2}\)의 부정적분 \(F(x)\)가 \(\displaystyle F(4)=\frac{5}{4}\)일 때, \(F(x)\)를 구하라.
풀이
\(\displaystyle F(x)=\int\left(\frac{1}{4}\sqrt{x}-\frac{1}{x^2}\right)dx=\int\left(\frac{1}{4}x^{\frac{1}{2}}-x^{-2}\right)dx\)
\(\displaystyle=\frac{1}{4}\cdot\frac{2}{3}x^{\frac{3}{2}}-(-x^{-1})+C=\frac{1}{6}\sqrt{x^3}+x^{-1}+C\)
\(\displaystyle=\frac{1}{6}x\sqrt{x}+\frac{1}{x}+C\)
\(\displaystyle F(4)=\frac{5}{4}=\frac{1}{6}4\sqrt{4}+\frac{1}{4}+C\)
\(\displaystyle C=\frac{5}{4}-\frac{1}{6}4\cdot2-\frac{1}{4}=1-\frac{4}{3}=-\frac{1}{3}\)
\(\displaystyle F(x)=\frac{1}{6}x\sqrt{x}+\frac{1}{x}-\frac{1}{3}\)