7.5 Integration by Parts

7.5 Integration by Parts

For two functions $u$ and $v$:

$$\int u\,v\,dx = u\int v\,dx - \int\left(\frac{du}{dx}\int v\,dx\right)dx$$

Use the ILATE rule to choose $u$ (pick the function that comes first):

• I — Inverse trigonometric
• L — Logarithmic
• A — Algebraic
• T — Trigonometric
• E — Exponential

Example

Evaluate $\displaystyle\int x\,e^x\,dx$.

Let $u = x$ (algebraic), $v = e^x$ (exponential).

$$\int x\,e^x\,dx = x \cdot e^x - \int 1 \cdot e^x\,dx = xe^x - e^x + C = e^x(x - 1) + C$$