$\int f\left(x\right)\cdot g\prime \left(x\right) \mathit{\ⅆ}x\=f\left(x\right)g\left(x\right)-\int f\prime \left(x\right)\cdot g\left(x\right) \mathit{\ⅆ}x$

$\int u \mathrm{dv}\=u v-\int v \mathit{\ⅆ}u$

${\int }_a^{b}f\left(x\right)\cdot g\prime \left(x\right) \mathit{\ⅆ}x\={\left[f\left(x\right)g\left(x\right)\right]}_{x\=a}^{x\=b}-{\int }_a^{b}f\prime \left(x\right)\cdot g\left(x\right) \mathit{\ⅆ}x$

${\int }_a^{b}u \mathit{\ⅆ}v\={\left[u v\right]}_a^b-{\int }_a^{b}v \mathit{\ⅆ}u$

Table 6.1.1   Various forms of the integration-by-parts algorithm

$\frac{d}{\mathrm{dx}}\left(u\left(x\right)\cdot v\left(x\right)\right)$

$\=u\prime \left(x\right)\cdot v\left(x\right)\+u\left(x\right)\cdot v\prime \left(x\right)$

$\int \frac{d}{\mathrm{dx}}\left(u\left(x\right)\cdot v\left(x\right)\right) \mathit{\ⅆ}x$

$\=\int u\prime \left(x\right)\cdot v\left(x\right) \mathit{\ⅆ}x\+\int u\left(x\right)\cdot v\prime \left(x\right) \mathit{\ⅆ}x$

$u\left(x\right)v\left(x\right)-\int u\prime \left(x\right)\cdot v\left(x\right) \mathit{\ⅆ}x$

$\=\int u\left(x\right)\cdot v\prime \left(x\right) \mathit{\ⅆ}x$

Example 6.1.1

Using the technique of integration by parts, evaluate $\int x \sin \left(x\right) \mathit{\ⅆ}x$.

Example 6.1.2

Using the technique of integration by parts, evaluate $\int \ln \left(x\right) \mathit{\ⅆ}x$.

Example 6.1.3

Using the technique of integration by parts, evaluate the definite integral ${\int }_0^1{\tan }^{-1}\left(x\right) \mathit{\ⅆ}x$.

Example 6.1.4

Use integration by parts to establish the formula

 $\int e^{a x}\cos \left(b x\right) \mathit{\ⅆ}x$ = $\frac{e^{a x}}{a^2\+b^2}\left(a \cos \left(b x\right)\+b \sin \left(b x\right)\right)$.

Example 6.1.5

Use integration by parts to evaluate the indefinite integral $\int x^2\sin \left(x\right) \mathit{\ⅆ}x$.