Questions: Evaluate the integral. (Remember to use absolute values where appropriate.) [ int0^6 frac4 t(t-7)^2 d t ]

$Evaluate the integral. (Remember to use absolute values where appropriate.) [ int0^6 frac4 t(t-7)^2 d t ]$

Transcript text: Evaluate the integral. (Remember to use absolute values where appropriate.) \[ \int_{0}^{6} \frac{4 t}{(t-7)^{2}} d t \]

Solution

Solution Steps

To evaluate the integral $\int_{0}^{6} \frac{4 t}{(t-7)^{2}} d t$, we can use a symbolic computation library in Python. The integral involves a rational function, which can be handled using symbolic integration. We will define the function, set up the integral with the given limits, and compute the result.

Step 1: Define the Integral

We are given the integral to evaluate: \[ \int_{0}^{6} \frac{4t}{(t-7)^{2}} \, dt \]

Step 2: Evaluate the Integral

The integral of the function $\frac{4t}{(t-7)^{2}}$ from 0 to 6 is calculated as: \[ \int \frac{4t}{(t-7)^{2}} \, dt = 24 - 4\ln|7| \]

Step 3: Simplify the Expression

Substitute the natural logarithm value to simplify the expression: \[ 24 - 4\ln|7| \]