Question Statement

Evaluate the integral:

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

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Background and Explanation

This problem requires the power rule for integration, which states that $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ for any real number $n\ne -1$. You'll also need to express square roots as fractional exponents to apply this rule.

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Solution

Let $I=\int \left(\sqrt{x}+\frac{1}{2\sqrt{x}}\right)dx$

First, we use the linearity property of integrals to split this into two separate integrals:

$I=\int \sqrt{x}dx+\frac{1}{2}\int \frac{1}{\sqrt{x}}dx$

Next, convert the radical expressions to fractional exponent form. Recall that $\sqrt{x}=x^{1/2}$ and $\frac{1}{\sqrt{x}}=x^{-1/2}$:

$I=\int x^{1/2}dx+\frac{1}{2}\int x^{-\frac{1}{2}}dx$

Now apply the power rule for integration $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ to each term:

• For the first term: $n=\frac{1}{2}$, so $n+1=\frac{3}{2}$
• For the second term: $n=-\frac{1}{2}$, so $n+1=\frac{1}{2}$

$I=\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\frac{1}{2}\left(\frac{x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\right)+c$

Simplify the exponents and denominators:

$I=\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+\frac{1}{2}\left(\frac{x^{\frac{1}{2}}}{\frac{1}{2}}\right)+c$

Simplify the fractions by dividing by the denominators (which is equivalent to multiplying by their reciprocals):

• $\frac{x^{3/2}}{3/2}=\frac{2}{3}{x}^{3/2}$
• $\frac{1}{2}\cdot \frac{x^{1/2}}{1/2}=\frac{1}{2}\cdot 2\cdot x^{1/2}=x^{1/2}$

Therefore:

$I=\frac{2}{3}{x}^{\frac{3}{2}}+x^{\frac{1}{2}}+c$

Or equivalently:

$I=\frac{2}{3}x\sqrt{x}+\sqrt{x}+c$

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Key Formulas or Methods Used

• Linearity of Integration: $\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$ and $\int k\cdot f(x)dx=k\int f(x)dx$ for constant $k$
• Exponent Conversion: $\sqrt{x}=x^{1/2}$ and $\frac{1}{\sqrt{x}}=x^{-1/2}$
• Power Rule for Integration: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ (for $n\ne -1$)

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Summary of Steps

1. Split the integral using linearity into $\int \sqrt{x}dx+\frac{1}{2}\int \frac{1}{\sqrt{x}}dx$
2. Convert to exponents rewriting $\sqrt{x}$ as $x^{1/2}$ and $\frac{1}{\sqrt{x}}$ as $x^{-1/2}$
3. Apply the power rule to each term: $\frac{x^{3/2}}{3/2}$ for the first and $\frac{1}{2}\cdot \frac{x^{1/2}}{1/2}$ for the second
4. Simplify coefficients: $\frac{1}{3/2}=\frac{2}{3}$ and $\frac{1}{2}\cdot \frac{1}{1/2}=1$
5. Combine results: $\frac{2}{3}{x}^{3/2}+x^{1/2}+C$