Answer :
Final answer:
The question involves the properties of definite integrals, particularly, linearity. After setting x to 0 in the simplified function, which is a common method when seeking f(0), the result doesn't match any of the provided options, suggesting there's an error in the question.
Explanation:
This is a question about an integral property known as linearity of integration, involving definite integrals and a function f(t).
The given equation can be simplified by subtracting ∫xˣ2f(t)dt on both sides, yielding ∫₀ˣ²f(t)dt - ∫ₓˣ²f(t)dt = −80x + 117eˣ.
By the property of linearity, this simplifies to ∫₀ˣf(t)dt = −80x + 117eˣ. The left side is the area under the graph of the function f(t), from 0 to x. However, we are looking for f(0), i.e., the value of the function at 0.
Noting that the integral of a function evaluated at 0 equals zero, we can set x to 0 in our equalised function, which simplifies to 0 = -80*0 + 117e^0, giving us 0 = 117.
However, there seems to be an error in the question, as none of the provided options match this result.
Learn more about Definite Integrals here:
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