Cyclic integrals and classical traps

The case of cyclic integrals

Some integrals, such as the integral of e^x cos(x) dx, can never be simplified completely: two successive application of the integration by parts (IPP) result in the original integral, multiplied by a factor. These are said to be cyclic.

Detailed example

Let C = the integral of e^x cos(x) dx. Substitution 1: u = cos(x), dv = e^x dx → du = -sin(x) dx, v = e^x. C = e^x cos(x) + the integral of e^x sin(x) dx. Substitution method 2 applied to the integral of e^x sin(x) dx: u = sin(x), dv = e^x dx → du = cos(x) dx, v = e^x. ∫ e^x sin(x) dx = e^x sin(x) – ∫ e^x cos(x) dx = e^x sin(x) – C.

Substituting back: C = e^x cos(x) + e^x sin(x) - C, so 2C = e^x (cos x + sin x), hence C = e^x (cos x + sin x) / 2 + K.

The method to remember

When the original integral reappears in the calculation, simply treat it as an algebraic unknown and solve the resulting equation.

Common pitfalls to avoid

• Changing the choice of u/dv between two successive integration steps: this cancels the cycle and causes the original integral to disappear rather than reappear.
• Forgetting the integration constant when drawing conclusions.
• Confusing definite and indefinite integrals: for a definite integral, the cycle is carried out over the antiderivative, and the limits are only evaluated at the end.
• Failing to check the result by differentiation, which would help detect a common sign error involving -sin(x).