Solved: "Evaluate the following integral. ∫(x+1x3)dx"

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zachutnat4o

Answered question

2021-11-22

Evaluate the following integral.
$\int (\sqrt{x} + \frac{1}{\sqrt[3]{x}}) d x$

Answer & Explanation

Warajected53

Beginner2021-11-23Added 12 answers

Step 1
The given integral is $\int (\sqrt{x} + \frac{1}{\sqrt[3]{x}}) d x$
Evaluate the above integral as follows.
Step 2
$\int (\sqrt{x} + \frac{1}{\sqrt[3]{x}}) d x = \int (x^{\frac{1}{2}} + \frac{1}{3} x^{- \frac{1}{2}}) d x$
$= \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + \frac{1}{3} \frac{x^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1} + C$
$= \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + \frac{1}{3} \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C$
$= \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{3} x^{\frac{1}{2}} + C$
$= \frac{2}{3} (x \sqrt{x} + \sqrt{x}) + C$
Thus, $\int (\sqrt{x} + \frac{1}{\sqrt[3]{x}}) d x = \frac{2}{3} (x \sqrt{x} + \sqrt{x}) + C$

Harr1957

Beginner2021-11-24Added 18 answers

Step 1: Expand.
$\int (\sqrt{x} + \frac{1}{\sqrt[3]{x}}) d x$
Step 2: Use Sum Rule: $\int f (x) + g (x) d x = \int f (x) d x + \int g (x) d x$ .
$\int \sqrt{x} d x + \int \frac{1}{\sqrt{3} {x}} d x$
Step 3: Since $\sqrt{x} = x^{\frac{1}{2}}$ , using the PowerRule, $\int x^{\frac{1}{2}} d x = \frac{2}{3} x^{\frac{3}{2}}$
$\frac{2 x^{\frac{3}{2}}}{3} + \int \frac{1}{\sqrt[3]{x}} d x$
Step 4: Use Power Rule: $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C$ .
$\frac{2 x^{\frac{3}{2}}}{3} + \frac{3}{2 x^{- \frac{2}{3}}}$
Step 5: Add constant.
$\frac{2 x^{\frac{3}{2}}}{3} + \frac{3}{2 x^{- \frac{2}{3}}} + C$

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