Questions: Directions: Calculate the following definite integral. [ int1^3 48 xleft(x^2-6right)^5 d x= ]

Transcript text: Directions: Calculate the following definite integral. \[ \int_{1}^{3} 48 x\left(x^{2}-6\right)^{5} d x= \] $\square$

Solution

Solution Steps

Step 1: Define the Integral

We start with the definite integral we want to evaluate: \[ \int_{1}^{3} 48 x\left(x^{2}-6\right)^{5} dx \]

Step 2: Substitution

Let $ u = x^2 - 6 $. Then, the differential $ du $ is given by: \[ du = 2x \, dx \quad \Rightarrow \quad dx = \frac{du}{2x} \] We also need to change the limits of integration. When $ x = 1 $: \[ u = 1^2 - 6 = -5 \] When $ x = 3 $: \[ u = 3^2 - 6 = 3 \]

Step 3: Rewrite the Integral

Substituting $ u $ and $ dx $ into the integral, we have: \[ \int_{-5}^{3} 48 x \left(u\right)^{5} \cdot \frac{du}{2x} = \int_{-5}^{3} 24 u^{5} du \]

Step 4: Evaluate the Integral

Now we evaluate the integral: \[ \int 24 u^{5} du = 24 \cdot \frac{u^{6}}{6} = 4 u^{6} \] We then evaluate this from $ u = -5 $ to $ u = 3 $: \[ 4 \left[ (3)^{6} - (-5)^{6} \right] \]

Step 5: Calculate the Result

Calculating the values: \[ (3)^{6} = 729 \quad \text{and} \quad (-5)^{6} = 15625 \] Thus, we have: \[ 4 \left[ 729 - 15625 \right] = 4 \cdot (-14996) = -59984 \]

Step 6: Final Result

The value of the definite integral is: \[ \int_{1}^{3} 48 x\left(x^{2}-6\right)^{5} dx = -59584 \]