∫ x^(2/3)√x dx

Steps to Solve

1. Rewrite the integrand

The integrand can be simplified. Since $\text{√} , x = x^{1/2}$, we can rewrite the integral:
$$ \int x^{2/3} \sqrt{x} , dx = \int x^{2/3} \cdot x^{1/2} , dx $$

2. Combine the exponents

Using the property of exponents, $a^m \cdot a^n = a^{m+n}$, we can combine the terms:
$$ = \int x^{2/3 + 1/2} , dx $$

3. Calculate the combined exponent

To combine $2/3$ and $1/2$, we need a common denominator:

• The common denominator of 3 and 2 is 6.
• Convert $2/3$ to $4/6$ and $1/2$ to $3/6$.
• Thus, $2/3 + 1/2 = \frac{4}{6}+\frac{3}{6}=\frac{7}{6}$.

So the integral now becomes:
$$ = \int x^{7/6} , dx $$

4. Integrate the function

We apply the power rule of integration, which states $\int x^n , dx = \frac{x^{n+1}}{n+1} + C$, where $C$ is the constant of integration.
Here, we have $n = 7/6$:
$$ = \frac{x^{7/6 + 1}}{7/6 + 1} + C $$

5. Calculate the new exponent and simplify

The new exponent is:
$$ 7/6 + 1 = 7/6 + 6/6 = 13/6 $$
Thus, we can write the integral as:
$$ = \frac{x^{13/6}}{13/6} + C $$
To simplify further:
$$ = \frac{6}{13} x^{13/6} + C $$