Question

Find a formula for the function represented by the integral .u3 du.2. Find the formula for the function represented by the integral.student submitted image, transcription available below(t + 8) dt

Answer 1

The formula for the function represented by the integral
`\( \int (t + 8) \, dt \) is \( (1)/(2)t^2 + 8t + C \)`, where ( C ) is the constant of integration.

Step-by-step explanation:

To find the formula for the function represented by the integral
`\( \int (t + 8) \, dt \),` we apply the rules of integration. The integral of ( t + 8 ) with respect to
`\( t \) is \( (1)/(2)t^2 + 8t + C \)`, where ( C ) is the constant of integration. The antiderivative of ( t ) is
`\( (1)/(2)t^2 \)`, and the antiderivative of 8 with respect to ( t ) is ( 8t ). Adding these antiderivatives and the constant of integration ( C ) gives us the final formula
`\( (1)/(2)t^2 + 8t + C \)`.

Understanding the process of integration is crucial in calculus. The antiderivative of a function represents the reverse process of differentiation, and the constant of integration accounts for the family of functions that differ only by a constant. In this case, the formula
`\( (1)/(2)t^2 + 8t + C \)` describes the set of functions whose derivative is ( t + 8 ).

In summary, the formula
`\( (1)/(2)t^2 + 8t + C \)` represents the function obtained by integrating ( t + 8 ) with respect to ( t ). The constant of integration ( C ) allows for the inclusion of all possible functions in the antiderivative family.