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An indefinite integral is a fundamental concept in calculus that represents the antiderivative of a function. It essentially answers the question: "What function, when differentiated, results in the given function?"

Key Concepts:

General Definition: The indefinite integral of a function f(x)f(x) is written as:

∫f(x) dx\int f(x) \, dx

It represents the collection of all functions whose derivative is f(x)f(x).

Constant of Integration: Since the derivative of a constant is zero, there can be infinitely many functions whose derivative is f(x)f(x). Therefore, the indefinite integral includes a constant of integration CC, which accounts for any constant added to the antiderivative:

∫f(x) dx=F(x)+C\int f(x) \, dx = F(x) + C

where F(x)F(x) is any antiderivative of f(x)f(x).

Antiderivative: An antiderivative of a function f(x)f(x) is a function F(x)F(x) such that:

F′(x)=f(x)F'(x) = f(x)

For example, if f(x)=2xf(x) = 2x, the antiderivative is F(x)=x2F(x) = x^2, and the indefinite integral is:

∫2x dx=x2+C\int 2x \, dx = x^2 + C

Basic Integration Rules: There are several standard rules for evaluating indefinite integrals:

Power Rule: For any real number n≠−1n \neq -1:

∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

Sum Rule: The integral of a sum is the sum of the integrals:

∫(f(x)+g(x)) dx=∫f(x) dx+∫g(x) dx\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx

Constant Multiple Rule: If kk is a constant:

∫k⋅f(x) dx=k⋅∫f(x) dx\int k \cdot f(x) \, dx = k \cdot \int f(x) \, dx

Exponential Rule: For exe^x, the antiderivative is simply:

∫ex dx=ex+C\int e^x \, dx = e^x + C

Trigonometric Integrals: Common integrals for trigonometric functions include:

∫sin⁡(x) dx=−cos⁡(x)+C,∫cos⁡(x) dx=sin⁡(x)+C\int \sin(x) \, dx = -\cos(x) + C, \quad \int \cos(x) \, dx = \sin(x) + C

Techniques of Integration: In more complex cases, specific techniques are used to evaluate indefinite integrals:

Substitution: This method is useful when the integrand is a composite function. You substitute a part of the function with a new variable to simplify the integral.

Integration by Parts: This technique is based on the product rule for differentiation. If you have two functions u(x)u(x) and v(x)v(x), you can use:

∫u(x)v′(x) dx=u(x)v(x)−∫v(x)u′(x) dx\int u(x) v'(x) \, dx = u(x) v(x) - \int v(x) u'(x) \, dx

Partial Fractions: This technique is often used when integrating rational functions, by expressing the function as a sum of simpler fractions.

Example of an Indefinite Integral:

Let’s compute the indefinite integral of f(x)=3x2f(x) = 3x^2:

∫3x2 dx\int 3x^2 \, dx

Using the power rule, we increase the exponent by 1 (from 2 to 3) and divide by the new exponent:

=3x33+C=x3+C= \frac{3x^3}{3} + C = x^3 + C

Thus, the indefinite integral is x3+Cx^3 + C.

In summary, an indefinite integral gives a family of functions (the antiderivatives) whose derivative is the integrand.

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