Integration by parts

This method comes from the product rule for derivatives. With u the first function and v the second: ∫u v dx = u∫v dx − ∫(u′ ∫v dx) dx.

In words: first function × integral of the second, minus the integral of (derivative of the first × integral of the second).

Choose as the first function the one that gets simpler when differentiated (a power of x, a logarithm, an inverse trigonometric function), and as the second one you can integrate easily.

Worked example: ∫x cos x dx with x first gives x sin x − ∫sin x dx = x sin x + cos x + C. Taking cos x first instead makes the new integral harder, not easier.

Worked example: ∫log x dx has no obvious second function, so take 1 as the second: x log x − ∫x(1/x) dx = x log x − x + C. The same trick handles ∫sin⁻¹x dx and ∫tan⁻¹x dx.

Worked example: ∫x eˣ dx = x eˣ − ∫eˣ dx = x eˣ − eˣ + C.

Returning integrals: in ∫eˣ sin x dx, applying the rule twice brings back the original integral. Moving it to one side gives ∫eˣ sin x dx = (eˣ/2)(sin x − cos x) + C.

No constant is needed when writing ∫g(x) dx inside the rule; any constant added there cancels out.

The method does not always help: for √x sin x there is no choice that makes the new integral simpler.