Integration by Parts & Partial Fractions

Integration by parts reverses the product rule. It turns a hard integral into a product term plus an easier integral.

You split the integrand into a part to differentiate ($u$) and a part to integrate ($dv$), then assemble $uv - \int v\,du$.

LIATE orders function types by how good a choice they are for $u$. Pick the one that appears earliest in the list.

• L. Logarithmic, like ln(x).
• I. Inverse trig, like arctan(x).
• A. Algebraic, like x or x².
• T. Trigonometric, like sin(x).
• E. Exponential, like eˣ.

When the algebraic factor is a power of x, each pass of parts drops that power by one. Repeat parts until the remaining integral is elementary, as with $\int x^{2} e^{x}\,dx$.

A rational function with distinct linear factors in the denominator splits into a sum of simpler fractions.

Clear the denominators to get a polynomial identity, then solve for the constants. The fastest route is to substitute the root of each factor, which zeroes out the other terms.

Once decomposed, each piece integrates to a logarithm or an arctangent.

This tool verifies each antiderivative numerically with Simpson's rule, so you can watch the worked answer match the area under the curve.