Integration by Parts: Repeated Application

At A-Level you learn the basic integration by parts formula. In Further Mathematics, you need to apply it repeatedly — sometimes two, three, or even more times — and handle cyclic integrals where the original integral reappears during the process. This lesson covers these advanced techniques, including the tabular method for speed.

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1 Review of Integration by Parts

The formula is:

integral of u (dv/dx) dx = uv - integral of v (du/dx) dx

Or in shorthand: integral of u dv = uv - integral of v du.

The key skill is choosing u and dv wisely. The LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) suggests a priority order for u.

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2 Repeated Integration by Parts

Sometimes one application of integration by parts does not finish the job — the resulting integral still requires parts. You simply apply the formula again.

Worked Example 1

Evaluate the integral of x^2 e^x dx.

Solution:

First application: u = x^2, dv = e^x dx. du = 2x dx, v = e^x.

integral of x^2 e^x dx = x^2 e^x - integral of 2x e^x dx