use a change of variables or the table of general integration formulas to evaluate the following definite…

use a change of variables or the table of general integration formulas to evaluate the following definite integral.\n int_{4}^{5}xsqrt{25 - x^{2}}dx \nclick to view the table of general integration formulas.\n int_{4}^{5}xsqrt{25 - x^{2}}dx=square \text{(type an exact answer.)}

use a change of variables or the table of general integration formulas to evaluate the following definite integral.\n int_{4}^{5}xsqrt{25 - x^{2}}dx \nclick to view the table of general integration formulas.\n int_{4}^{5}xsqrt{25 - x^{2}}dx=square \text{(type an exact answer.)}

Answer

Explanation:

Step1: Let (u = 25 - x^{2})

Differentiate (u) with respect to (x): (du=-2x\ dx), so (x\ dx=-\frac{1}{2}du). When (x = 4), (u=25 - 16 = 9). When (x = 5), (u=25 - 25=0).

Step2: Rewrite the integral

The integral (\int_{4}^{5}x\sqrt{25 - x^{2}}dx) becomes (-\frac{1}{2}\int_{9}^{0}\sqrt{u}du).

Step3: Integrate (\sqrt{u})

We know that (\int u^{n}du=\frac{u^{n + 1}}{n+1}+C) for (n\neq - 1). Here (n=\frac{1}{2}), so (\int\sqrt{u}du=\int u^{\frac{1}{2}}du=\frac{2}{3}u^{\frac{3}{2}}+C).

Step4: Evaluate the definite - integral

(-\frac{1}{2}\int_{9}^{0}\sqrt{u}du=-\frac{1}{2}\left[\frac{2}{3}u^{\frac{3}{2}}\right]_{9}^{0}). [ \begin{align*} &=-\frac{1}{2}\times\frac{2}{3}(0 - 9^{\frac{3}{2}})\ &=-\frac{1}{3}(0 - 27)\ & = 9 \end{align*} ]

Answer:

9