Math Snap

STEP 1

1. The integral $\int_{4}^{5} x \sqrt{x^{2}-16} \, dx$ is a definite integral.
2. We will use substitution to simplify the integral.
3. The limits of integration will need to be adjusted according to the substitution.

STEP 2

1. Choose an appropriate substitution to simplify the integrand.
2. Change the limits of integration according to the substitution.
3. Perform the integration.
4. Evaluate the definite integral.

STEP 3

Choose a substitution. Let $u = x^2 - 16$. Then, differentiate $u$ with respect to $x$:
$$\frac{du}{dx} = 2x$$ $$du = 2x \, dx$$ Solve for $x \, dx$:
$$x \, dx = \frac{1}{2} du$$

STEP 4

Change the limits of integration. When $x = 4$, calculate $u$:
$$u = 4^2 - 16 = 0$$ When $x = 5$, calculate $u$:
$$u = 5^2 - 16 = 9$$ Thus, the new limits of integration are from $u = 0$ to $u = 9$.

STEP 5

Substitute into the integral:
$$\int_{4}^{5} x \sqrt{x^{2}-16} \, dx = \int_{0}^{9} \sqrt{u} \cdot \frac{1}{2} \, du$$ Simplify the integral:
$$\frac{1}{2} \int_{0}^{9} u^{1/2} \, du$$

STEP 6

Perform the integration:
$$\frac{1}{2} \int u^{1/2} \, du = \frac{1}{2} \left[ \frac{u^{3/2}}{3/2} \right]_{0}^{9}$$ $$= \frac{1}{2} \cdot \frac{2}{3} \left[ u^{3/2} \right]_{0}^{9}$$ $$= \frac{1}{3} \left[ u^{3/2} \right]_{0}^{9}$$

Evaluate the definite integral:
$$= \frac{1}{3} \left[ 9^{3/2} - 0^{3/2} \right]$$ Calculate $9^{3/2}$:
$$9^{3/2} = (9^{1/2})^3 = 3^3 = 27$$ So, the integral evaluates to:
$$= \frac{1}{3} \times 27 = 9$$ The value of the definite integral is:
$$\boxed{9}$$