Integration by Parts

Understanding Integration by Parts

Integration by parts is a powerful technique for solving integrals, rooted in the product rule for differentiation. It's especially useful when dealing with products of functions.

The Formula

The integration by parts formula is given by:

∫ u dv = uv - ∫ v du

Where:

• u is a function of x.
• dv is another function of x.

Choosing the appropriate u and dv is crucial for simplifying the problem.

Example Problem

Let's consider ∫ x e^x dx:

• Choose u = x, thus du = dx.
• Choose dv = e^x dx, thus v = e^x.

Applying the formula gives us:

∫ x e^x dx = x e^x - ∫ e^x dx = x e^x - e^x + C

Tips for Using Integration by Parts

• When choosing u and dv, consider functions that simplify when differentiated or integrated.
• The LIATE rule (Logarithmic, Inverse, Algebraic, Trigonometric, Exponential) can guide the choice of u.

Want to try more exercises? Visit our practice section for more problems and solutions.