To solve the definite integral \(\int_4^7 x \sqrt{x-2} \, dx\), we can follow these steps:
1. Integrand Identification: The integrand is \( x \sqrt{x-2} \).
2. Substitution: To simplify the integral, we use the substitution method. Let \( u = x - 2 \). Therefore, \( du = dx \) and \( x = u + 2 \).
3. Changing the Limits of Integration: The original limits are from \( x = 4 \) to \( x = 7 \). We need to convert these to \( u \)-values:
- When \( x = 4 \), \( u = 4 - 2 = 2 \).
- When \( x = 7 \), \( u = 7 - 2 = 5 \).
So, the new limits of integration in terms of \( u \) are from \( 2 \) to \( 5 \).
4. Substituting \( x \):
Substitute \( x = u + 2 \) into the integrand:
[tex]\[
x \sqrt{x-2} = (u + 2) \sqrt{u}
\][/tex]
5. Simplifying the Integrand:
The integrand now becomes:
[tex]\[
(u + 2) \sqrt{u} = u \sqrt{u} + 2 \sqrt{u}
= u^{3/2} + 2u^{1/2}
\][/tex]
6. Integral in Terms of \( u \):
The integral is now:
[tex]\[
\int_{2}^{5} \left( u^{3/2} + 2u^{1/2} \right) du
\][/tex]
7. Evaluating the Integral:
We can integrate each term separately:
[tex]\[
\int_{2}^{5} u^{3/2} \, du + 2 \int_{2}^{5} u^{1/2} \, du
\][/tex]
First integral:
[tex]\[
\int u^{3/2} \, du = \frac{2}{5} u^{5/2}
\][/tex]
Evaluating from 2 to 5:
[tex]\[
\left[ \frac{2}{5} u^{5/2} \right]_{2}^{5} = \frac{2}{5} \left[ 5^{5/2} - 2^{5/2} \right]
\][/tex]
Second integral:
[tex]\[
2 \int u^{1/2} \, du = 2 \left( \frac{2}{3} u^{3/2} \right) = \frac{4}{3} u^{3/2}
\][/tex]
Evaluating from 2 to 5:
[tex]\[
\left[ \frac{4}{3} u^{3/2} \right]_{2}^{5} = \frac{4}{3} \left[ 5^{3/2} - 2^{3/2} \right]
\][/tex]
8. Combining the Results:
Summing the evaluated integrals:
[tex]\[
\frac{2}{5} \left( 5^{5/2} - 2^{5/2} \right) + \frac{4}{3} \left( 5^{3/2} - 2^{3/2} \right)
\][/tex]
9. Final Numerical Result:
After plugging in the values and simplifying, we obtain the result:
[tex]\[
-\frac{64\sqrt{2}}{15} + \frac{50\sqrt{5}}{3}
\][/tex]
Thus, the value of the definite integral [tex]\(\int_4^7 x \sqrt{x-2} \, dx\)[/tex] is [tex]\(-\frac{64\sqrt{2}}{15} + \frac{50\sqrt{5}}{3}\)[/tex].
