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Antiderivatives

Antiderivatives, which are also referred to as indefinite integrals or primitive functions, is essentially the opposite of a derivative (hence the name). More formally, an antiderivative F is a function whose derivative is equivalent to the original function f, or stated more concisely: F′(x)=f(x).

The Fundamental Theorem of Calculus defines the relationship between differential and integral calculus. We will see later that an antiderivative can be thought of as a restatement of an indefinite integral. Therefore, the discussion of antiderivatives provides a nice segue from the differential to integral calculus.

Finding Antiderivatives

The process of finding an antiderivative of a function is known as antidifferentiation and is the reverse of differentiating a function.

For example, consider the function f(x)=x3. The antiderivative of this function, F(x), is 14x4+C. As antidifferentiation is the reverse operation of differentiation, we solve for the antiderivative by finding the original function before the differentiation took place. A constant C, known as the constant of integration, is added to the antiderivative as the derivative of any constant is 0. The addition of the constant does not affect the correspondence of the original antiderivative.

The general antiderivative of f(x)=xn, valid when n≥0, is:

F(x)=xn+1n+1+C
In [2]:
from sympy import symbols, limit, diff, sin, cos, log, tan, sqrt, init_printing, plot, integrate
from mpmath import ln, e, pi

init_printing()
x = symbols('x')
y = symbols('y')

We can confirm the antiderivative of the function f(x)=x3 is 14x4+C with SymPy using the integrate() function.

In [3]:
integrate(x ** 3, x)
Out[3]:
x44

Examples

Example 1: Find the antiderivative of f(x)=x−3

Using the general antiderivative form above:

F(x)=12x2−3x+C
In [13]:
integrate(x - 3, x)
Out[13]:
x22−3x

Example 2: Determine the antiderivative of the function f(x)=12+34x2−45x3

Going left to right and performing the reverse of the differentiation operation:

F(x)=12x+14x4−15x4+C
In [14]:
integrate(x / 2 + (3 / 4) * x ** 2 - (4 / 5) * x ** 3, x)
Out[14]:
−0.2x4+0.25x3+0.25x2

Example 3: Find the antiderivative of the function f(x)=(x+1)(2x−1)

Expanding the function then performing antidifferentiation:

f(x)=2x2−x−1F(x)=23x3+12x2−x+C
In [15]:
integrate((x + 1) * (2 * x - 1), x)
Out[15]:
2x33+x22−x

Example 4: Find the antiderivative of the function f(x)=10x9

Rewriting the function as f(x)=101x9 and antidifferentiating:

F(x)=−108x−8=−54x−8
In [18]:
integrate(10 / x ** 9, x)
Out[18]:
−54x8

Example 5: Find the antiderivative of g(θ)=cosθ−5sinθ

Antidifferentiate each piece of the function:

G(θ)=sinθ+5cosθ
In [19]:
integrate(cos(x) - 5 * sin(x))
Out[19]:
sin(x)+5cos(x)

References

Antiderivative. (2017, December 29). In Wikipedia, The Free Encyclopedia. From https://en.wikipedia.org/w/index.php?title=Antiderivative&oldid=817529505

Stewart, J. (2007). Essential calculus: Early transcendentals. Belmont, CA: Thomson Higher Education.

Weisstein, Eric W. "Constant of Integration." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ConstantofIntegration.html

Weisstein, Eric W. "Indefinite Integral." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IndefiniteIntegral.html

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