Froge Calculus Tutorials

Integration by Parts

Integration by parts is an important technique in the integration toolbox. It is based on the product rule for differentiation and can be expressed as:

∫ u dv = uv - ∫ v du

Here's how it's typically applied:

1. Identify the parts of the integrand: choose which part to differentiate and which to integrate (usually u and dv respectively).
2. Differentiate u to find du and integrate dv to find v.
3. Plug into the integration by parts formula.

Example

Let's consider the integral ∫ x e^x dx:

1. Choose u = x → du = dx, dv = e^x dx → v = e^x
2. Apply the formula: ∫ x e^x dx = x e^x - ∫ e^x dx
3. Simplify: = x e^x - e^x + C

This technique is particularly useful for integrals of the form ∫ xⁿ e^ax dx, ∫ xⁿ sin(ax) dx, and ∫ xⁿ cos(ax) dx.

Keep hopping through mathematics with Froge!