Question

$\int \left(x{)}^{\frac{1}{3}}\right(2+x^{\frac{1}{2}}{)}^{2}dx$

• A. $7x^{\frac{4}{3}}+\frac{3}{7}{x}^{\frac{7}{3}}+\frac{24}{11}{x}^{\frac{11}{6}}+C$
• B. $4x^{\frac{4}{3}}+\frac{3}{7}{x}^{\frac{7}{3}}+\frac{24}{11}{x}^{\frac{11}{6}}+C$
• C. $3x^{\frac{4}{3}}+\frac{3}{7}{x}^{\frac{7}{3}}+\frac{24}{11}{x}^{\frac{11}{6}}+C$
• D. $11x^{\frac{4}{3}}+\frac{3}{7}{x}^{\frac{7}{3}}+\frac{24}{11}{x}^{\frac{11}{6}}+C$

Answer 1

The correct option is C $3x^{\frac{4}{3}}+\frac{3}{7}{x}^{\frac{7}{3}}+\frac{24}{11}{x}^{\frac{11}{6}}+C$

The given problem is of $\int \left(x{)}^m\right(a+b{x}^n{)}^p$ dx form with “ p” being a natural number. To solve such problems we’ll expand $(a+b{x}^n{)}^p$ using binomial and integrate. $\int \left(x{)}^{\frac{1}{3}}\right(2+x^{\frac{1}{2}}{)}^{2}dx=\int \left(x{)}^{\frac{1}{3}}\right(4+x+4x^{\frac{1}{2}})dx=\int 4(x{)}^{\frac{1}{3}}dx+\int x^{\frac{4}{3}}dx+\int 4x^{\frac{5}{6}}dx=\frac{4x^{\frac{4}{3}}}{\frac{4}{3}}+\frac{x^{\frac{7}{3}}}{\frac{7}{3}}+\frac{4x^{\frac{11}{6}}}{\frac{11}{6}}+C=3x^{\frac{4}{3}}+\frac{3}{7}{x}^{\frac{7}{3}}+\frac{24}{11}{x}^{\frac{11}{6}}+C$