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Section, 5.2 Integration by parts of the chapter, 5. Techniques of integration.

• Reversing the Product Rule: If $u$and $v$are functions such that $u\text{'}\left(x\right)v\left(x\right)+u\left(x\right)v\text{'}\left(x\right)$is integrable, then $鈭�\left(u\text{'}\left(x\right)v\left(x\right)+u\left(x\right)v\text{'}\left(x\right)\right)dx=u\left(x\right)v\left(x\right)+C$
• The formula for integration by parts: If $u=u\left(x\right)$and $v=v\left(x\right)$are differentiable functions, then $鈭�udv=uv-鈭�vdu$
• It is best to try algebraic simplification and u-substitution before attempting integration by parts.
• Choose $u$and $dv$so that the associated $du$and $v$are simpler than what we started with.
• Integral of the Natural Logarithm Function:$鈭�\ln xdx=x\ln x-x+C$
• Integrals of Inverse Sine and Inverse Tangent: $$\begin{gathered} 鈭�{\sin }^{-1}xdx=x{\sin }^{-1}x+\sqrt{1-x^2}+C \\ 鈭�{\tan }^{-1}xdx=x{\tan }^{-1}x-\frac{\ln \left(x^2+1\right)}{2}+C \end{gathered}$$
• Integration by Parts for Definite Integrals: If $u=u\left(x\right)$and $v=v\left(x\right)$are differentiable functions on $\left[a,b\right]$, then ${鈭�}_a^{b}udv={\left[鈭�udv\right]}_a^b={\left[uv\right]}_a^b-{鈭�}_a^{b}vdu$