Implementing Integrals in Python
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Computing an Indefinite Integral (Antiderivative)
An indefinite integral represents the antiderivative of a function. It finds the general form of a function whose derivative gives the original function.
1234567891011import sympy as sp # Define function x = sp.Symbol('x') f = x**2 # Compute indefinite integral F = sp.integrate(f, x) # Output: x**3 / 3 print(F)
Computing a Definite Integral (Area Under Curve)
A definite integral finds the accumulated sum of a function over a range [a,b].
1234567891011121314import sympy as sp # Define function x = sp.Symbol('x') f = x**2 # Define integration limits a, b = 0, 2 # Compute definite integral integral_value = sp.integrate(f, (x, a, b)) # Output: 8/3 print(integral_value)
Common Integrals in Python
Python allows us to compute common mathematical integrals symbolically. Here are a few examples:
123456789101112131415161718import sympy as sp # Define function x = sp.Symbol('x') # Exponential integral exp_integral = sp.integrate(sp.exp(x), x) # Sigmoid function integral sigmoid_integral = sp.integrate(1 / (1 + sp.exp(-x)), x) # Quadratic function integral quadratic_integral = sp.integrate(2*x, (x, 0, 2)) # Print results print(exp_integral) # Output: e^x print(sigmoid_integral) # Output: log(1 + e^x) print(quadratic_integral) # Output: 4
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