Final Answer
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Simplify $\sqrt[3]{x^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{3}$
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$\int\left(\sqrt{x}-2\sqrt[3]{x^{2}}\right)\left(x-\sqrt{x}\right)dx$
Learn how to solve problems step by step online. Integrate int((x^1/2-2x^2^1/3)(x-x^1/2))dx. Simplify \sqrt[3]{x^2} using the power of a power property: \left(a^m\right)^n=a^{m\cdot n}. In the expression, m equals 2 and n equals \frac{1}{3}. Rewrite the integrand \left(\sqrt{x}-2\sqrt[3]{x^{2}}\right)\left(x-\sqrt{x}\right) in expanded form. Expand the integral \int\left(\sqrt{x^{3}}-x-2\sqrt[3]{x^{5}}+2\sqrt[6]{x^{7}}\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\sqrt{x^{3}}dx results in: \frac{2}{5}\sqrt{x^{5}}.