Integration by parts

Objective. Integrate products using a deliberate choice of u.

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Learn: the key idea

Integration by parts handles the integral of a product of two functions using the formula ∫u(dv/dx)dx = uv − ∫v(du/dx)dx. Choose u as the factor that simplifies on differentiation (such as x or ln x) and dv/dx as the factor that is easy to integrate. For ∫ x e^x dx, let u = x and dv/dx = e^x, so du/dx = 1 and v = e^x, giving xe^x − ∫e^x dx = xe^x − e^x + c. A sensible choice of u makes the remaining integral genuinely simpler.

Worked example

Use integration by parts to find ∫ x cos x dx.

1 $Let u = x and dv/dx = cos x, so du/dx = 1 and v = sin x$
2 $Apply the formula: x sin x - \int sin x dx$
3 $Integrate the remaining term: -\intsin x dx = cos x$
4 $Combine to give x sin x + cos x + c$

Answer: x sin x + cos x + c

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Common mistake. Choosing u = e^x and dv/dx = x when integrating xe^x makes the resulting integral more complicated rather than simpler. Always pick u so that differentiating it reduces its degree, leaving an easier integral.

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