Answer:
22
Step-by-step explanation:
Answer:
See Explanation.
General Formulas and Concepts:
<u>Pre-Algebra</u>
- Distributive Property
- Equality Properties
<u>Algebra I</u>
- Combining Like Terms
- Factoring
<u>Calculus</u>
- Derivative 1:
![\frac{d}{dx} [e^u]=u'e^u](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%20%5Be%5Eu%5D%3Du%27e%5Eu)
- Integration Constant C
- Integral 1:

- Integral 2:

- Integral 3:

- Integral Rule 1:

- Integration by Parts:

- [IBP] LIPET: Logs, Inverses, Polynomials, Exponents, Trig
Step-by-step Explanation:
<u>Step 1: Define Integral</u>

<u>Step 2: Identify Variables Pt. 1</u>
<em>Using LIPET, we determine the variables for IBP.</em>
<em>Use Int Rules 2 + 3.</em>

<u>Step 3: Integrate Pt. 1</u>
- Integrate [IBP]:

- Integrate [Int Rule 1]:

<u>Step 4: Identify Variables Pt. 2</u>
<em>Using LIPET, we determine the variables for the 2nd IBP.</em>
<em>Use Int Rules 2 + 3.</em>

<u>Step 5: Integrate Pt. 2</u>
- Integrate [IBP]:

- Integrate [Int Rule 1]:

<u>Step 6: Integrate Pt. 3</u>
- Integrate [Alg - Back substitute]:
![\int {e^{au}sin(bu)} \, du = \frac{-e^{au}cos(bu)}{b} + \frac{a}{b} [\frac{e^{au}sin(bu)}{b} - \frac{a}{b} \int ({e^{au} sin(bu)}) \, du]](https://tex.z-dn.net/?f=%5Cint%20%7Be%5E%7Bau%7Dsin%28bu%29%7D%20%5C%2C%20du%20%3D%20%5Cfrac%7B-e%5E%7Bau%7Dcos%28bu%29%7D%7Bb%7D%20%2B%20%5Cfrac%7Ba%7D%7Bb%7D%20%5B%5Cfrac%7Be%5E%7Bau%7Dsin%28bu%29%7D%7Bb%7D%20-%20%5Cfrac%7Ba%7D%7Bb%7D%20%5Cint%20%28%7Be%5E%7Bau%7D%20sin%28bu%29%7D%29%20%5C%2C%20du%5D)
- [Integral - Alg] Distribute Brackets:

- [Integral - Alg] Isolate Original Terms:

- [Integral - Alg] Rewrite:

- [Integral - Alg] Isolate Original:

- [Integral - Alg] Rewrite Fraction:

- [Integral - Alg] Combine Like Terms:

- [Integral - Alg] Divide:

- [Integral - Alg] Multiply:
![\int {e^{au}sin(bu)} \, du = \frac{1}{a^2+b^2} [ae^{au}sin(bu) - be^{au}cos(bu)]](https://tex.z-dn.net/?f=%5Cint%20%7Be%5E%7Bau%7Dsin%28bu%29%7D%20%5C%2C%20du%20%3D%20%5Cfrac%7B1%7D%7Ba%5E2%2Bb%5E2%7D%20%5Bae%5E%7Bau%7Dsin%28bu%29%20-%20be%5E%7Bau%7Dcos%28bu%29%5D)
- [Integral - Alg] Factor:
![\int {e^{au}sin(bu)} \, du = \frac{e^{au}}{a^2+b^2} [asin(bu) - bcos(bu)]](https://tex.z-dn.net/?f=%5Cint%20%7Be%5E%7Bau%7Dsin%28bu%29%7D%20%5C%2C%20du%20%3D%20%5Cfrac%7Be%5E%7Bau%7D%7D%7Ba%5E2%2Bb%5E2%7D%20%5Basin%28bu%29%20-%20bcos%28bu%29%5D)
- [Integral] Integration Constant:
![\int {e^{au}sin(bu)} \, du = \frac{e^{au}}{a^2+b^2} [asin(bu) - bcos(bu)] + C](https://tex.z-dn.net/?f=%5Cint%20%7Be%5E%7Bau%7Dsin%28bu%29%7D%20%5C%2C%20du%20%3D%20%5Cfrac%7Be%5E%7Bau%7D%7D%7Ba%5E2%2Bb%5E2%7D%20%5Basin%28bu%29%20-%20bcos%28bu%29%5D%20%2B%20C)
And we have proved the integration formula!
Answer:
<u>158 natural numbers from 78 to 234, and 699 whole numbers from 24 to 721.</u>
Step-by-step explanation:
Natural numbers are positive integers (whole numbers), so all numbers from the range of 78 to 234 would be included, including 78 and 234 itself. That is 158 natural numbers. Whole numbers are numbers without a fraction or decimal, they are integers. So all numbers from 24 to 721 would be included, as well as 24 and 721 itself. That is 699 whole numbers from 24 to 721. Hope this helps.