Integration by parts

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

Choose u so that its derivative simplifies the problem and dv so that its primitive is accessible. Compute du and v before substituting. LIATE is a guideline: logarithms, inverse trigonometric, algebraic, trigonometric and exponential functions. It does not guarantee the best choice in every case.

Worked example

1. Identify the integrand, variables and order. For multiple integrals, start with the innermost integral.

   $$\int x e^{x}\, dx$$

2. Integrate with respect to x. Treat the other variables as constants at this stage.

   $$\int x e^{x}\, dx$$

3. Choose u and dv. Differentiate u and find a primitive v of dv.

   $$u=x,\quad dv=e^{x}\,dx,\quad du=1\,dx,\quad v=e^{x}$$

4. Substitute into the integration by parts formula. The minus sign applies to the entire remaining integral.

   $$\int x e^{x}\, dx = x e^{x} - \int e^{x}\, dx$$

5. The exponential is its own derivative. Compensate any constant factor in the exponent.

   $$\int e^{x}\, dx = e^{x}$$

6. Check this primitive by differentiating it: recover exactly the integrand of this stage.

   $$\left(x - 1\right) e^{x} + e^{x} = x e^{x}$$

7. Add the constant C: all primitives on an interval differ by a constant.

   $$\left(x - 1\right) e^{x} + C$$