How to know when to use integration by substitution vs. integration by parts?

Here's a plain text explanation to help you decide when to use substitution versus integration by parts:

Integration by Substitution:

Use substitution when the integral contains a function and its derivative (or something close to it). This method simplifies the integral by changing variables.

Typical scenarios:

You notice a composite function, such as f(g(x)), and g'(x) (or a multiple of it) is present.
Examples:
∫sin(2x) dx → Substitution: Let u = 2x.
∫x e^(x^2) dx → Substitution: Let u = x^2.
Steps:

Identify a substitution (e.g., set u = g(x)).
Replace dx and other expressions in terms of u.
Solve the integral in terms of u.
Substitute back the original variable.
Integration by Parts:

Use this method when the integral involves a product of two functions and substitution doesn’t simplify it effectively. It relies on the formula: ∫u dv = uv - ∫v du.

Typical scenarios:

One function is easily differentiable (u), and the other is easy to integrate (dv).
Examples:
∫x e^x dx → Use integration by parts: Let u = x, dv = e^x dx.
∫ln(x) dx → Rewrite as ∫1·ln(x) dx and apply integration by parts.
Steps:

Choose u and dv based on the LIATE rule (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential).
Differentiate u to find du, and integrate dv to find v.
Substitute into the formula ∫u dv = uv - ∫v du.
Simplify and solve the resulting integral.
General Guideline:

Start by checking if substitution can simplify the integral. It is often quicker and works for many standard problems.
If substitution does not help and the integral involves a product of functions, consider integration by parts.
Examples for Comparison:

Substitution: ∫(2x + 1)^3 dx → Substitution: Let u = 2x + 1.
By Parts: ∫x sin(x) dx → Integration by parts: Let u = x, dv = sin(x) dx.
By practicing problems, you’ll develop intuition for choosing the right method.