Calculus

Study of continuous change through derivatives and integrals.

Advanced Topics

Techniques of Integration

Integrating More Complex Functions

Not all integrals are straightforward! Sometimes you need special tricks to solve them.

Substitution

Change variables to make the integral easier, like undoing the chain rule in reverse.

Integration by Parts

Used when you have a product of two functions. It's based on the product rule for derivatives.

\[ \int u,dv = uv - \int v,du \]

Partial Fractions

Break complicated fractions into simpler pieces you can integrate.

Why Learn These?

These techniques let you solve a wider range of real-world problems involving areas, volumes, and accumulations.

Key Formula

\[\int u,dv = uv - \int v,du\]

Examples

  • Using substitution to find \( \int 2x \cos(x^2) dx \).

  • Integrating \( \int x e^x dx \) by parts.

In a Nutshell

Advanced methods help solve tricky integrals for more complex scenarios.

Key Terms

Substitution
A method of changing variables to simplify integration.
Integration by Parts
A technique for integrating products of functions.