Question

Math

Posted 7 months ago

`S(n) is a statement about positive integers n such that whenever S(k) is true, S(k + 1) must also be true. Furthermore, there exists some positive integer n0 such that S(n0) is not true. Of the following, which is the strongest conclusion that can be drawn?`

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Answer from Sia

Posted 7 months ago

Solution by Steps

step 1

To find the definite integral of $x^4$ from $x = 3$ to $x = 5$, we use the fundamental theorem of calculus

step 2

The antiderivative of $x^4$ is $\frac{x^5}{5}$ plus a constant

step 3

We evaluate $\frac{x^5}{5}$ at the upper limit of integration $x = 5$ and at the lower limit $x = 3$

step 4

The definite integral is $\left[\frac{5^5}{5}\right] - \left[\frac{3^5}{5}\right]$

step 5

Simplifying, we get $\frac{3125}{5} - \frac{243}{5}$

step 6

The result is $\frac{2882}{5}$, which can also be written as 576.4

Answer

The definite integral of $x^4$ from $x = 3$ to $x = 5$ is $\frac{2882}{5}$ or 576.4.

Key Concept

Definite Integral of a Polynomial Function

Explanation

The definite integral of a polynomial function $x^n$ from $a$ to $b$ is found by evaluating the antiderivative $\frac{x^{n+1}}{n+1}$ at $b$ and $a$ and then subtracting the two results.

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