To solve the integral:

$I = \int_0^1 x \left( 10x^{1/3} + 7x^{1/4} \right) \, dx$

We first break the integral into two parts:

$I = \int_0^1 10x^{1/3 + 1} \, dx + \int_0^1 7x^{1/4 + 1} \, dx$

This simplifies to:

$I = \int_0^1 10x^{4/3} \, dx + \int_0^1 7x^{5/4} \, dx$

Now, integrate each term:

1. First term:

$\int_0^1 10x^{4/3} \, dx = 10 \cdot \frac{x^{(4/3) + 1}}{(4/3) + 1} \bigg|_0^1 = 10 \cdot \frac{x^{7/3}}{7/3} \bigg|_0^1 = 10 \cdot \frac{3}{7} \left[ x^{7/3} \right]_0^1$

Evaluating this:

$= 10 \cdot \frac{3}{7} (1^{7/3} - 0^{7/3}) = 10 \cdot \frac{3}{7} = \frac{30}{7}$

2. Second term:

$\int_0^1 7x^{5/4} \, dx = 7 \cdot \frac{x^{(5/4) + 1}}{(5/4) + 1} \bigg|_0^1 = 7 \cdot \frac{x^{9/4}}{9/4} \bigg|_0^1 = 7 \cdot \frac{4}{9} \left[ x^{9/4} \right]_0^1$

Evaluating this:

$= 7 \cdot \frac{4}{9} (1^{9/4} - 0^{9/4}) = 7 \cdot \frac{4}{9} = \frac{28}{9}$

Total value of the integral:

$I = \frac{30}{7} + \frac{28}{9}$

To add these fractions, find the common denominator:

$I = \frac{30}{7} + \frac{28}{9} = \frac{30 \cdot 9}{63} + \frac{28 \cdot 7}{63} = \frac{270 + 196}{63} = \frac{466}{63}$

Thus, the value of the integral is:

$I = \frac{466}{63}$

Would you like a more detailed explanation or further questions on integrals? Here are some related questions you might find interesting:

1. What is the process to integrate polynomial expressions in general?
2. How do you handle improper integrals or integrals with infinite limits?
3. Can you integrate rational functions using partial fractions?
4. What is the power rule for integration, and how is it applied to fractional exponents?
5. How does integration by parts work, and when is it useful?

Tip: When dealing with fractional powers, always carefully adjust the exponents during integration. Ensure you apply the power rule correctly for fractions.