Unit 4: Substitution Rule of Indefinite Integrals

The Substitution Rule, also known as the u-substitution method, is a pivotal technique in calculus for simplifying the integration of complex functions, especially composite function [latex]f(g(x))[/latex]. It involves substituting a new variable, typically denoted as [latex]u[/latex], to transform the composite function as [latex]f(u)[/latex], where [latex]u=g(x)[/latex], and streamline the integration process.

Based on the Chain Rule for differentiation, the Substitution Rule leverages the relationship between $[latex]u[/latex]$ and $[latex]x[/latex]$ to reconfigure the integral. If [latex]\mathbf{\mathcal{u=g(x)}}[/latex] is a differentiable function of [latex]x[/latex], and [latex]f(u)[/latex] is continuous on the range of [latex]u[/latex], then the rule states that

[latex]\int f(g(x)) g'(x) dx =\int f(u)du[/latex]

By rewriting the integral in terms of $[latex]u[/latex]$, we can often simplify the expression and make it more manageable.

The substitution rule of definite integral can be found at Unit 9.

Practice Questions:

Evaluate the following indefinite integrals before referring to the video for comprehensive solutions.

Q1. [latex]\int (1-2x)^5 dx[/latex]

Q2. [latex]\int x\sqrt{x^2-3}\; dx[/latex]

Q3. [latex]\int \cos(7\theta-5)  \;d\theta[/latex]

Q4. [latex]\int x^2 e^{x^3-1} \;dx[/latex]

Q5. [latex]\int \frac{4}{z \ln^2 z} \;dz[/latex]

Q6. [latex]\int t \sqrt{t+2} \;dt[/latex]

Q7. [latex]\int \frac{3^{1/y}}{y^2} \;dy[/latex]