Substitution Rule For Indefinite Integrals

Yaser Rahmati | یاسر رحمتی

Substitution Rule

Here is the substitution rule in general.

$$\int f(g(x))g'(x) dx = \int f(u) du$$

Example

Let’s start with the one example.

$$\int 18x^2 \sqrt{6x^3 + 5} dx$$

In this case let’s notice that if we let

$$u=6x^3+5$$

and we compute the differential for this we get,

$$du=18x^3$$

Now, let’s go back to our integral and notice that we can eliminate every $x$ that exists in the integral and write the integral completely in terms of $u$ using both the definition of $u$ and its differential.

$$\int 18x^2 \sqrt{6x^3 + 5} dx=\int \sqrt{u}du =\frac{2}{3}u^\frac{3}{2} + c$$

Evaluating the integral gives,

$$F(x)=(6x^3+5)^\frac{3}{2}+c$$

Python

import sympy as sp

# Define the symbolic variable
x = sp.symbols('x')

# Define the function symbolically
f = 18*(x**2)*sp.Pow(6*x**3 + 5,0.5)

# Calculate the integral of the function
F = sp.integrate(f, x)

# Print the result
print('The integral of the function f(x) is:',F)

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