Evaluate the integral ∫ x^(2/3) √x dx.

Steps to Solve

1. Rewrite the Integral

First, we will rewrite the integral in a simpler form. The square root function can be represented as an exponent.

The integral becomes:

$$ \int x^{2/3} x^{1/2} , dx $$

2. Combine the Exponents

Next, we need to combine the exponents. We can use the property of exponents that states $a^m a^n = a^{m+n}$.

Adding the exponents:

$$ \int x^{2/3 + 1/2} , dx $$

To add ( \frac{2}{3} ) and ( \frac{1}{2} ), we find a common denominator:

$$ \frac{2}{3} = \frac{4}{6}, \quad \frac{1}{2} = \frac{3}{6} $$

Thus,

$$ 2/3 + 1/2 = \frac{4}{6} + \frac{3}{6} = \frac{7}{6} $$

So we have:

$$ \int x^{7/6} , dx $$

3. Integrate

Now we can find the antiderivative using the power rule for integration, which states:

$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$

Applying the rule here:

$$ \int x^{7/6} , dx = \frac{x^{7/6 + 1}}{7/6 + 1} + C $$

First, calculate ( 7/6 + 1 ):

$$ 7/6 + 1 = \frac{7}{6} + \frac{6}{6} = \frac{13}{6} $$

Thus, the integral becomes:

$$ \frac{x^{13/6}}{13/6} + C $$

This can be simplified further:

$$ = \frac{6}{13} x^{13/6} + C $$