Evaluate the following integrals:
Question:

Evaluate the following integrals:

$\int \frac{x^{-1 / 3}+\sqrt{x}+2}{\sqrt[3]{x}} d x$

Solution:

Given:

$\int \frac{x^{\frac{1}{a}}+\sqrt{x}+2}{\sqrt[3]{x}} d x$

By Splitting them,

$\Rightarrow \int \frac{x^{-\frac{1}{3}}}{\sqrt[2]{x}} d x+\int \frac{\sqrt{x}}{\sqrt[3]{x}} d x+\int \frac{2}{\sqrt[3]{x}} d x$

$\Rightarrow \int x^{-\frac{1}{3}} \times x^{-\frac{1}{3}} d x+\int x^{\frac{1}{2}} \times x^{-\frac{1}{3}} d x+2 \int x^{-\frac{1}{3}} d x$

$\Rightarrow \int x^{-\frac{1}{2}-\frac{1}{3}} d x+\int x^{\frac{1}{2}-\frac{1}{3}} d x+2 \int x^{-\frac{1}{2}} d x$

$\Rightarrow \int x^{-\frac{2}{3}} d x+\int x^{\frac{5}{6}} d x+2 \int x^{-\frac{1}{3}} d x$

By applying the formula,

$\int x^{n} d x=\frac{x^{n+1}}{n+1}$

We get,

$\Rightarrow \frac{x^{-\frac{2}{3}+1}}{-\frac{2}{3}+1}+\frac{x^{\frac{5}{6}+1}}{\frac{5}{6}+1}+\frac{2 x^{-\frac{1}{2}+1}}{-\frac{1}{3}+1}+c$

$\Rightarrow \frac{x^{\frac{1}{2}}}{\frac{1}{3}}+\frac{x^{\frac{11}{6}}}{\frac{11}{6}}+\frac{2 x^{\frac{2}{2}}}{\frac{2}{3}}+c$

$\Rightarrow 3 x^{\frac{1}{2}}+\frac{6 x^{\frac{11}{6}}}{11}+3 x^{\frac{2}{3}}+c$

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