Solve ∫ 3left(sqrt{x}-sqrt[3]{x^2}+sqrt[5]{2}right) wrt x

$$\int{ 3 \left( \sqrt{ x } - \sqrt[ 3 ]{ { x }^{ 2 } } + \sqrt[ 5 ]{ 2 } \right) }d x$$

Evaluate

$$-\frac{9x^{\frac{5}{3}}}{5}+2x^{\frac{3}{2}}+3\sqrt[5]{2}x+С$$

Solution Steps

Use the distributive property to multiply 3 by \sqrt{x}-\sqrt[3]{x^{2}}+\sqrt[5]{2}.

$$\int 3\left(\sqrt{x}-\sqrt[3]{x^{2}}\right)+3\sqrt[5]{2}\mathrm{d}x$$

Use the distributive property to multiply 3 by \sqrt{x}-\sqrt[3]{x^{2}}.

$$\int 3\sqrt{x}-3\sqrt[3]{x^{2}}+3\sqrt[5]{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3\sqrt{x}\mathrm{d}x+\int -3x^{\frac{2}{3}}\mathrm{d}x+\int 3\sqrt[5]{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int \sqrt{x}\mathrm{d}x-3\int x^{\frac{2}{3}}\mathrm{d}x+3\int \sqrt[5]{2}\mathrm{d}x$$

Rewrite \sqrt{x} as x^{\frac{1}{2}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{2}}\mathrm{d}x with \frac{x^{\frac{3}{2}}}{\frac{3}{2}}. Simplify. Multiply 3 times \frac{2x^{\frac{3}{2}}}{3}.

$$2x^{\frac{3}{2}}-3\int x^{\frac{2}{3}}\mathrm{d}x+3\int \sqrt[5]{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{2}{3}}\mathrm{d}x with \frac{3x^{\frac{5}{3}}}{5}. Multiply -3 times \frac{3x^{\frac{5}{3}}}{5}.

$$2x^{\frac{3}{2}}-\frac{9x^{\frac{5}{3}}}{5}+3\int \sqrt[5]{2}\mathrm{d}x$$

Find the integral of \sqrt[5]{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$2x^{\frac{3}{2}}-\frac{9x^{\frac{5}{3}}}{5}+3\sqrt[5]{2}x$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$2x^{\frac{3}{2}}-\frac{9x^{\frac{5}{3}}}{5}+3\sqrt[5]{2}x+С$$

Differentiate w.r.t. x

$$3\left(-x^{\frac{2}{3}}+\sqrt{x}+\sqrt[5]{2}\right)$$