Evaluate \displaystyle \int_1^{10} \left(\sqrt{x} + 1\right)^3 dx - \int_1^{10} \left(\sqrt{x} - 1\right)^3 dx .
Hint:
For x > 0, you have
(sqrt(x) + 1)^3 - (sqrt(x) - 1)^3 =
6 x + 2
To evaluate the given expression, we need to calculate two definite integrals and then subtract the value of one from the other.
Let's start by simplifying the expression inside the first integral:
\((\sqrt{x} + 1)^3 = \left(\sqrt{x} + 1\right)\left(\sqrt{x} + 1\right)\left(\sqrt{x} + 1\right) = (\sqrt{x})^3 + 3(\sqrt{x})^2 + 3\sqrt{x} + 1\).
Similarly, simplifying the expression inside the second integral:
\((\sqrt{x} - 1)^3 = \left(\sqrt{x} - 1\right)\left(\sqrt{x} - 1\right)\left(\sqrt{x} - 1\right) = (\sqrt{x})^3 - 3(\sqrt{x})^2 + 3\sqrt{x} - 1\).
Now, we can evaluate the definite integrals one by one.
For the first integral, \(\int_1^{10} \left(\sqrt{x} + 1\right)^3 dx\):
We can expand the expression and then integrate each term separately:
\((\sqrt{x})^3 + 3(\sqrt{x})^2 + 3\sqrt{x} + 1\)
\(= x^{3/2} + 3x + 3x^{1/2} + 1\).
Integrating each term:
\(\int_1^{10} x^{3/2} dx = \left[\frac{2}{5}x^{5/2}\right]_1^{10} = \frac{2}{5}\left(10^{5/2} - 1^{5/2}\right)\),
\(\int_1^{10} 3x dx = \left[\frac{3}{2}x^2\right]_1^{10} = \frac{3}{2}(10^2 - 1^2)\),
\(\int_1^{10} 3x^{1/2} dx = \left[2x^{3/2}\right]_1^{10} = 2(10^{3/2} - 1^{3/2})\),
\(\int_1^{10} 1 dx = \left[x\right]_1^{10} = 10 - 1\).
Combining all the integrals:
\(\int_1^{10} \left(\sqrt{x} + 1\right)^3 dx = \frac{2}{5}\left(10^{5/2} - 1^{5/2}\right) + \frac{3}{2}(10^2 - 1^2) + 2(10^{3/2} - 1^{3/2}) + (10 -1)\).
Now, let's evaluate the second integral, \(\int_1^{10} \left(\sqrt{x} - 1\right)^3 dx\):
By following the same steps as before, we can simplify and integrate this integral as well.
After evaluating both integrals, we can subtract the value of the second integral from the first to find the final answer.