Frequently Asked Questions: Integration Methods (Substitution & By Parts)
Integration by Substitution, often called U-Substitution, is an integration technique that reverses the chain rule from differentiation. It's used to simplify integrals where the integrand is a composite function multiplied by the derivative of the inner function. You substitute a new variable (commonly 'u') for a part of the integrand, transforming the integral into a simpler form that can be evaluated directly.
The general steps are:
- Choose a part of the integrand to be your 'u' (often the inner function of a composite).
- Calculate the differential du/dx and solve for dx in terms of du (e.g., dx = du / (du/dx)).
- Substitute 'u' and 'dx' into the original integral. The goal is for the entire integral to be in terms of 'u' and 'du'.
- Integrate the simplified expression with respect to 'u'.
- Substitute back the original expression for 'u' to get the result in terms of the original variable.
- (For definite integrals) Change the limits of integration to be in terms of 'u' instead of 'x', or integrate and then substitute back the original limits.
Choosing 'u' is often the trickiest part. Look for:
- The inner function of a composite function (e.g., u = x²+1 in ∫ 2x cos(x²+1) dx).
- An expression whose derivative is also present in the integrand (possibly multiplied by a constant).
- An expression under a root or in the denominator.
Integration by Parts is an integration technique that reverses the product rule from differentiation. It's used to integrate products of functions, particularly when a simple substitution doesn't work. The formula is: ∫ u dv = uv − ∫ v du. You choose one part of the integrand to be 'u' and the other part (including dx) to be 'dv'.
The main difference lies in the rule they reverse and when they are typically applied:
- **Substitution (U-Substitution):** Reverses the **Chain Rule**. Best used when the integrand is a composite function and you see the derivative of the inner function (or a multiple of it) present. It simplifies the integral by replacing the variable.
- **By Parts:** Reverses the **Product Rule**. Best used when the integrand is a product of two different types of functions (e.g., polynomial times exponential, x sin(x)) where substitution doesn't simplify the integral enough. It transforms the integral into a potentially easier one (∫ v du).
Start by looking for patterns suggestive of **U-Substitution**: Do you see a function and its derivative (or a constant multiple)? Is there an inner function whose derivative is present outside? (e.g., ex² · x, cos(3x) · 3, √(x+1) · 1).
If substitution doesn't seem to fit or doesn't simplify the integral, consider **Integration by Parts**. This is often useful for products of functions from different "families" like:
- Polynomial × Exponential (e.g., x ex)
- Polynomial × Trigonometric (e.g., x sin(x))
- Exponential × Trigonometric (e.g., ex cos(x))
- Logarithmic × Polynomial (e.g., ln(x) · x)
- Inverse Trigonometric × Polynomial (e.g., arctan(x) · x)
