Bài 2: Nguyên hàm của một số hàm số sơ cấp

1) \(\int\left(e^{x-2}-\dfrac{2}{sin^2x}\right)dx=\int e^{x-2}dx-\int\dfrac{2}{sin^2x}dx=e^{x-2}+2cotx+C\)

2) \(f\left(x\right)=\sqrt{x}+\sqrt[3]{x}+\sqrt[4]{x}=x^{\dfrac{1}{2}}+x^{\dfrac{1}{3}}+x^{\dfrac{1}{4}}\)

\(\int f\left(x\right)dx=\int x^{\dfrac{1}{2}}dx+\int x^{\dfrac{1}{3}}dx+\int x^{\dfrac{1}{4}}dx\)

\(=\dfrac{x^{\dfrac{1}{2}+1}}{\dfrac{1}{2}+1}+\dfrac{x^{\dfrac{1}{3}+1}}{\dfrac{1}{3}+1}+\dfrac{x^{\dfrac{1}{4}+1}}{\dfrac{1}{4}+1}+C=\dfrac{2x^{\dfrac{3}{2}}}{3}+\dfrac{3x^{\dfrac{4}{3}}}{4}+\dfrac{4x^{\dfrac{5}{4}}}{5}+C\)

3) \(\int f\left(x\right)dx=\int\left(\dfrac{1}{\sqrt{x}}-\dfrac{2}{\sqrt[3]{x}}\right)dx=\int x^{-\dfrac{1}{2}}dx-\int2x^{-\dfrac{1}{3}}dx\)

\(=\dfrac{x^{-\dfrac{1}{2}+1}}{-\dfrac{1}{2}+1}-\dfrac{2x^{-\dfrac{1}{3}+1}}{-\dfrac{1}{3}+1}+C=2x^{\dfrac{1}{2}}-3x^{\dfrac{2}{3}}+C\)

4) \(f\left(x\right)=\dfrac{\left(x+2\right)^2}{x^4}=\dfrac{x^2+4x+4}{x^4}=\dfrac{1}{x^2}+\dfrac{4}{x^3}+\dfrac{4}{x^4}=x^{-2}+4x^{-3}+4x^{-4}\)

\(\int f\left(x\right)dx=\int\left(x^{-2}+4x^{-3}+4x^{-4}\right)dx=\int x^{-2}dx+\int4x^{-3}dx+\int4x^{-4}dx\)

\(=\dfrac{x^{-2+1}}{-2+1}+\dfrac{4x^{-3+1}}{-3+1}+\dfrac{4x^{-4+1}}{-4+1}+C\)

\(=-x^{-1}-8x^{-2}-12x^{-3}+C\)

𝕙𝕒𝕖𝕟𝕘𝟚𝟘𝟙𝟘

\(f\left(x\right)=\sqrt{x\sqrt{x\sqrt{x}}}\)

\(\Leftrightarrow f\left(x\right)=\sqrt{x.x^{\dfrac{1}{2}}.x^{\dfrac{1}{4}}}=\sqrt{x^{\dfrac{7}{4}}}=x^{\dfrac{7}{8}}\)

\(\int f\left(x\right)dx=\int x^{\dfrac{7}{8}}dx=\dfrac{x^{\dfrac{7}{8}+1}}{\dfrac{7}{8}+1}+C=\dfrac{8x^{\dfrac{15}{8}}}{15}+C\)