The integral shown in the image is presented as follows:\[ \int \frac{dt}{t^2 - t - 6} \]The expression represents an indefinite integral where the integrand is the reciprocal of a quadratic polynomial. To solve this integral, one common approach is to use partial fraction decomposition. Here is a step-by-step outline of the procedure:1. **Factor the Denominator:** The quadratic polynomial in the denominator can be factored as: \[t^2 - t - 6 = (t - 3)(t + 2)\]2. **Set Up Partial Fractions:** \[ \frac{1}{t^2 - t - 6} = \frac{1}{(t - 3)(t + 2)} \] Express this as a sum of partial fractions: \[ \frac{1}{(t - 3)(t + 2)} = \frac{A}{t - 3} + \frac{B}{t + 2} \] where \(A\) and \(B\) are constants to be determined.3. **Solve for Constants:** \[ 1 = A(t + 2) + B(t - 3) \] By solving this equation for \(A\) and \(B\), we can find the values of the constants.4. **Integrate Each Term:** Once the constants are found, the integral can be written as the sum of easier integrals: \[ \int \left( \frac{A}{t - 3} + \frac{B}{t + 2} \right) dt \] Integrate each term separately to find the solution.For further detailed explanation and steps to solve the integral, you might want to refer to additional resources on partial fraction decomposition and integration methods.

Name: The integral shown in the image is presented as follows:\[ \int \frac
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: The integral shown in the image is presented as follows:\[ \int \frac{dt}{t^2 - t - 6} \]The expression represents an indefinite integral where the integrand is the reciprocal of a quadratic polynomial. To solve this integral, one common approach is