Repeated Integration by Parts

Calculus 2 · Axiom Academy

LESSON Repeated Integration by Parts When one application isn't enough: reducing powers, the tabular shortcut, and integrals that loop back on themselves. Recall the integration by parts formula — it trades one integral for a boundary term plus a new integral: For something like , one pass is not enough. Choosing u = x^2 knocks the power down to x , but it leaves behind — which still needs parts. The watch the animation below tracks: every application lowers the exponent by one until it hits x^0 . For a polynomial times an exponential or trig function, the tabular method ("D–I" tabular integration) does all those passes at once. Instead of writing out parts repeatedly, you organize the derivatives and integrals into a single table. Differentiate x^2 down the left ( ); integrate e^x down the right (it stays e^x ). Connect the diagonals with signs +, -, + and read off the products. 4. Circular Integration Patterns A second family behaves completely differently. Apply parts twice to and you do not simplify away the integral — instead the original integral reappears on the right-hand side. The same thing happens for . The table would never terminate, because neither e^x nor ever differentiates to 0 . Let . Apply parts twice (keeping e^x as the part you integrate both times). The integral returns with a coefficient, and a single algebra move solves for I .

This is the written version of the interactive lesson above. See the full Calculus 2 course.