draw a table to show how to solve the integral of functions

Here's a general table showing different methods to solve the integral of various types of functions:

| **Type of Function**                        | **Integral**                                | **Method**                              | **Example** |
|---------------------------------------------|---------------------------------------------|-----------------------------------------|-------------|
| **Constant Function**                       | ∫ c dx = cx + C                             | Basic integration                       | ∫ 3 dx = 3x + C |
| **Power of x (Polynomial)**                 | ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, (n ≠ -1)       | Power rule                              | ∫ x² dx = x³/3 + C |
| **Exponential Function**                    | ∫ eˣ dx = eˣ + C                           | Exponential rule                        | ∫ e²ˣ dx = e²ˣ/2 + C |
| **Trigonometric Functions (sin, cos, etc.)**| ∫ sin(x) dx = -cos(x) + C                   | Trigonometric integrals                 | ∫ cos(x) dx = sin(x) + C |
| **Logarithmic Function**                    | ∫ (1/x) dx = ln|x| + C                      | Logarithmic integral                    | ∫ (1/x) dx = ln|x| + C |
| **Integration by Substitution**             | ∫ f(g(x))g'(x) dx = ∫ f(u) du               | Substitution rule                       | ∫ 2x(x²+1) dx = (x²+1)²/2 + C |
| **Integration by Parts**                    | ∫ u dv = uv - ∫ v du                       | Integration by parts formula            | ∫ x eˣ dx = x eˣ - ∫ eˣ dx |
| **Partial Fractions (Rational Functions)**  | ∫ (P(x)/Q(x)) dx                           | Decompose into simpler fractions        | ∫ (1/(x²-1)) dx = ln|x-1| - ln|x+1| + C |
| **Trigonometric Substitution**              | ∫ sqrt(a² - x²) dx                         | Use trigonometric identities            | ∫ sqrt(1 - x²) dx = arcsin(x) + C |

### Example of a Method: Power Rule
For a polynomial function like \( x^3 \), the integral can be computed using the power rule:
\[
\int x^3 dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C
\]

This table outlines basic methods used to handle a variety of integrals, depending on the form of the function. The specific method to apply depends on the structure of the function you're integrating.

Do you want more details on a specific method?

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### 5 Related Questions:
1. How do you decide which integration method to use?
2. Can substitution work for all functions, or are there limitations?
3. What happens if you integrate a piecewise function?
4. How do you approach improper integrals where limits tend to infinity?
5. How do you handle integrals involving products of trigonometric functions?

### Tip:
When in doubt about which method to use, consider simplifying the integrand first. For example, using substitution or simplifying the algebraic form can make the integral easier to compute.

Learn various integration techniques, such as the power rule, trigonometric integrals, substitution, and integration by parts, with examples. This guide is perfect for college-level students studying calculus.

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