Copying the question word for word:
Investigate different rules for integration. You should include polynomials, exponential functions, logarithmic functions, sine functions, cosine functions and a function made up by a quotient.
Give an example of integrating each type of function listed above, and solve each example. Make sure each example you show is unique. Do not do more than ten examples.
Little stuck on this question. Hope someone can help me. Thanks!
Where are you getting stuck on this? Sounds like you just need to integrate a number of different mathematical statements.
To investigate different rules for integration, you can start by understanding the basic rules of integration and then applying them to each type of function listed in the question. Here's a step-by-step guide on how to approach each type of function and provide an example:
1. Polynomials:
- Example: Integrate f(x) = 3x^2 + 2x + 1.
- Solution: To integrate a polynomial term by term, use the power rule.
∫(3x^2) dx = x^3 + C1, ∫(2x) dx = x^2 + C2, ∫(1) dx = x + C3.
The final result is f(x) = x^3 + x^2 + x + C.
2. Exponential functions:
- Example: Integrate f(x) = e^x.
- Solution: The integral of e^x is the function itself.
∫e^x dx = e^x + C.
The final result is f(x) = e^x + C.
3. Logarithmic functions:
- Example: Integrate f(x) = ln(x).
- Solution: The integral of ln(x) involves using integration by parts.
∫ln(x) dx = x ln(x) - ∫(x / x) dx.
Simplify: ∫ln(x) dx = x ln(x) - ∫(1) dx = x ln(x) - x + C.
The final result is f(x) = x ln(x) - x + C.
4. Sine functions:
- Example: Integrate f(x) = sin(x).
- Solution: The integral of sin(x) is negative cosine.
∫sin(x) dx = -cos(x) + C.
The final result is f(x) = -cos(x) + C.
5. Cosine functions:
- Example: Integrate f(x) = cos(x).
- Solution: The integral of cos(x) is sine.
∫cos(x) dx = sin(x) + C.
The final result is f(x) = sin(x) + C.
6. Quotient function:
- Example: Integrate f(x) = (2x + 1) / x^2.
- Solution: The integral of a quotient function involves using partial fractions or long division.
First, rewrite f(x) as: f(x) = 2(x^(-1)) + x^(-2).
∫f(x) dx = ∫(2(x^(-1)) + x^(-2)) dx = 2∫(x^(-1)) dx + ∫(x^(-2)) dx.
Apply the power rule: 2ln|x| - x^(-1) + C.
The final result is f(x) = 2ln|x| - x^(-1) + C.
You can follow a similar approach for the remaining examples, ensuring that you apply the appropriate integration rules for each type of function. Remember to simplify your final results and include the constant of integration (C) since integration introduces an arbitrary constant.