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5.4.1 Indefinite Integration

Antiderivatives

An antiderivative of a function f(x) is a function F(x) such that the derivative of F(x) is f(x). In mathematical terms, if F'(x) = f(x), then F(x) is an antiderivative of f(x). It's crucial to understand that there are infinitely many antiderivatives for a given function, differing only by a constant. This is because differentiation of a constant results in zero, and hence, the original constant gets lost in the process. For instance, the antiderivative of sin(x) is -cos(x) + C, where C is an arbitrary constant.

Example:

Find the antiderivative of sin(x).

Solution:

The integral of sin(x) with respect to x is: ∫ sin(x) dx = -cos(x) + C Here, C is the constant of integration, representing any constant value.

Basic Formulas

Integration has a vast array of formulas, but here are some of the foundational ones that every maths student should know:

  • Power Rule: For any real number n not equal to -1, the integral of xn is: ∫ xn dx = (x(n+1))/(n+1) + C
  • Exponential Rule: The integral of the natural exponential function is: ∫ ex dx = ex + C
  • General Exponential Rule: For any positive number a different from 1, the integral of ax is: ∫ ax dx = (ax)/ln(a) + C
  • Logarithmic Rule: The integral of the natural logarithm is: ∫ ln(x) dx = x(ln(x) - 1) + C
  • Reciprocal Rule: The integral of the reciprocal function is: ∫ 1/x dx = ln|x| + C

Applications of Indefinite Integration

Indefinite integration isn't just a theoretical concept; it has practical applications in various fields of science, engineering, economics, and more.

Physics:

In physics, integration is used to find quantities like displacement when velocity is known, or electric potential when electric field is given.

Economics:

In economics, integration can be used to find consumer and producer surplus, calculate total cost from marginal cost, and more.

Biology:

In biology, integration can help in understanding growth rates, modelling population dynamics, and studying biological processes over time.

Example:

A car's velocity is given by v(t) = t2 + 2t. Find the car's displacement from time t = 0 to t = 2.

Solution:

To determine the displacement, we need to integrate the velocity function concerning time from 0 to 2.

s(t) = ∫ v(t) dt s(t) = ∫ (t2 + 2t) dt s(t) = (t3)/3 + t2 + C

Using the initial condition s(0) = 0 (assuming the car starts from the origin), we find C = 0.

Now, s(t) = (t3)/3 + t2.

To find the displacement from t = 0 to t = 2, evaluate s(2) - s(0).

s(2) = 8/3 + 4 = 20/3

Thus, the car's displacement in the given time interval is 20/3 units.

FAQ

Indefinite integration and definite integration are both forms of integration, but they serve different purposes. Indefinite integration finds the antiderivative of a function, resulting in a family of functions (due to the constant of integration). It does not have any bounds or limits. On the other hand, definite integration computes the area under a curve between two specific points (limits or bounds). The result of a definite integral is a numerical value, not a function. In essence, while indefinite integration gives a general form, definite integration provides a specific value.

Not all functions have elementary antiderivatives, meaning they cannot be expressed in terms of elementary functions like polynomials, exponentials, logarithms, trigonometric functions, and their inverses. Some functions require special methods or techniques for integration, while others might not have an antiderivative that can be expressed in a closed form. In such cases, numerical methods or approximations might be used, or the function might be expressed in terms of non-elementary functions like the error function (erf) or the Fresnel integrals.

The geometric interpretation of indefinite integration relates to the area under a curve. When we integrate a function, we are essentially finding the area between the function's graph and the x-axis. However, with indefinite integration, we don't have specific bounds, so we're looking at a general accumulation of area as we move along the x-axis. The constant of integration can be thought of as adjusting where we start accumulating this area. It's worth noting that while this interpretation is useful, especially in definite integration, indefinite integration is more about finding a function whose rate of change (derivative) is the given function.

Integration, especially indefinite integration, has numerous real-world applications across various fields. In physics, it's used to find quantities like displacement from velocity or electric potential from electric field. In biology, it can model population growth or the spread of diseases. In economics, integration can determine consumer surplus or total cost from a marginal cost function. Essentially, whenever there's a need to accumulate quantities or find totals based on rates of change, integration comes into play. It's a fundamental tool in understanding and modelling various phenomena in the real world.

The constant of integration represents an arbitrary constant that arises because differentiation removes any constant present in the original function. When we differentiate a function, any constant term becomes zero. Therefore, when we reverse the process through integration, we cannot determine the exact value of that constant. As a result, we add a general constant, often denoted as C, to represent all possible constants that could have been in the original function. This ensures that the family of functions represented by the antiderivative includes the original function.

Practice Questions

Evaluate the indefinite integral of the function f(x) = 3x^2 - 4x + 5.

To evaluate the indefinite integral of f(x) = 3x2 - 4x + 5, we'll integrate each term separately.

For the term 3x2: Integral of 3x2 dx = x3 + C1

For the term -4x: Integral of -4x dx = -2x2 + C2

For the constant term 5: Integral of 5 dx = 5x + C3

Combining all the integrals, we get: Integral of (3x2 - 4x + 5) dx = x3 - 2x2 + 5x + C Where C = C1 + C2 + C3 is the constant of integration.

Find the antiderivative of the function g(x) = e^x - ln(x).

To find the antiderivative of g(x) = ex - ln(x), we'll integrate each term separately.

For the term ex: Integral of ex dx = ex + D1

For the term -ln(x): Integral of -ln(x) dx = -xln(x) + x + D2

Combining the two integrals, we get: Integral of (ex - ln(x)) dx = ex - xln(x) + x + D Where D = D1 + D2 is the constant of integration.


Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
LinkedIn
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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