a.
\(\int\dfrac{1}{\sqrt{5x}-\sqrt{3x}}dx=\int\dfrac{\sqrt{5x}+\sqrt{3x}}{2x}dx\)
\(=\int\left(\dfrac{\sqrt{5x}}{2x}+\dfrac{\sqrt{3x}}{2x}\right)dx=\int\left(\dfrac{\sqrt{5}}{2.\sqrt{x}}+\dfrac{\sqrt{3}}{2\sqrt{x}}\right)dx\)
\(=\sqrt{5x}+\sqrt{3x}+C\)
b.
\(\int\left(x^2-1\right)^3dx=\int\left(x^6-3x^4+3x^2-1\right)dx\)
\(=\dfrac{1}{7}x^7-\dfrac{3}{5}x^5+x^3-x+C\)
c.
\(\int\left(2-x^2\right)^4dx=\int\left(x^8-8x^6+24x^4-32x^2+16\right)dx\)
\(=\dfrac{1}{9}x^9-\dfrac{8}{7}x^7+\dfrac{24}{5}x^5-\dfrac{32}{3}x^3+16x+C\)
d,
\(\int\left(\dfrac{x^2+2\sqrt[3]{x}}{x}\right)^2dx=\int\dfrac{x^4+4x^2\sqrt[3]{x}+4\sqrt[3]{x^2}}{x^2}dx\)
\(=\int\left(x^2+4\sqrt[3]{x}+\dfrac{4\sqrt[3]{x^2}}{x^2}\right)dx\)
\(=\int \left(x^2+4x^{\frac{1}{3}}+4x^{-\frac{4}{3}}\right)dx\)
\(=\frac{1}{3}x^3+\frac{4}{\frac{1}{3}+1}x^{\frac{1}{3}+1}+\frac{4}{-\frac{4}{3}+1}.x^{-\frac{4}{3}+1}+C\)
\(=\dfrac{1}{3}x^3+3\sqrt[3]{x^4}-\dfrac{12}{\sqrt[3]{x}}+C\)
