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)