### Indefinite Integral

**Concepts to remember for Indefinite Integral**

**1. Integration is the inverse process of differentiation.**

- In integration, we start with a function and find a new function whose derivative is the original function.

**2. The indefinite integral of a function is a function whose derivative is the given function.**

- The indefinite integral of a function (f(x)) is another function (F(x)) such that (F’(x) = f(x)).

**3. The indefinite integral of (f(x)) is denoted by (\int f(x) dx).**

- The symbol (∫) is used to denote the indefinite integral. The expression (\int f(x) dx) is read as “the integral of (f(x)) with respect to (x).”

**4. The general antiderivative of a function is the indefinite integral of the function plus a constant.**

- The general antiderivative of a function (f(x)) is the function (F(x) = \int f(x) dx + C), where (C) is a constant.

**5. The constant of integration is a real number that is added to the indefinite integral to obtain a definite integral.**

- The constant of integration allows us to find a specific function that is the antiderivative of a given function.

**6. The indefinite integral of a sum of two or more functions is the sum of the indefinite integrals of each function.**

- (\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx)

**7. The indefinite integral of a constant times a function is the constant times the indefinite integral of the function.**

- (\int c f(x) dx = c \int f(x) dx)

**8. The indefinite integral of a product of two functions is not equal to the product of the indefinite integrals of the two functions.**

- (\int f(x) g(x) dx ≠ \int f(x) dx \int g(x) dx)

**9. The indefinite integral of a quotient of two functions is not equal to the quotient of the indefinite integrals of the two functions.**

- (\int \frac{f(x)}{g(x)} dx ≠ \frac{\int f(x) dx}{\int g(x) dx})

**10. The indefinite integral of a power function is the power function with the exponent increased by 1, divided by the new exponent.**

- (\int x^n dx = \frac{x^{n+1}}{n+1} + C), where (n ≠ -1)

**11. The indefinite integral of an exponential function is the exponential function multiplied by the natural logarithm of the base of the exponential function.**

- (\int e^x dx = e^x + C)

**12. The indefinite integral of a logarithmic function is the logarithmic function multiplied by the natural logarithm of the argument of the logarithmic function.**

- (\int \ln x dx = x \ln x - x + C)

**13. The indefinite integral of a trigonometric function can be found using the following formulas:**

- (\int \sin x dx = -\cos x + C)
- (\int \cos x dx = \sin x + C)
- (\int \tan x dx = \ln |\sec x| + C)
- (\int \cot x dx = \ln |\sin x| + C)
- (\int \sec x dx = \ln |\sec x + \tan x| + C)
- (\int \csc x dx = -\ln |\csc x + \cot x| + C)