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Mathematics LibreTexts

1.1: Integration by parts

  • Page ID
    10264
  • [ "stage:draft", "article:topic", "Integration by Parts", "authorname:thangarajahp" ]

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    Integration by parts (IBP)

    \[\int\,u \,dv= u\,v-\int\, v\, du.\]

    Note

    When deciding which function should be u and dv, it is best to choose u as the function that will simplify easily.

    The priority for choosing u is:

    1. \(u = ln x\)

    2. \(u = x^n \),  where n= integer

    3. \( u = e^{nx}\), where n= integer

    Example \(\PageIndex{1}\):

    Find \( \int x^2 \, e^{-x}\, dx \).

    Answer:

    Using the priority listed above, let \(u=x^2\) and \(dv= e^{-x} \,dx\).

    Since \(u=x^2 \)  ,              \(du=2x \) .

    Since   \(dv= e^{-x} \),        \(v= \int e^{-x} \, dx).

                                                \(= -e^{-x}\)

    Using the IBP, 

    \(\int\,u \,dv= u\,v-\int\, v\, du\)

                   \(= x^2\, e^{-x}\ -\int\, e^{-x}\, 2x\, dx\)

                    \(= ln x\, \dfrac{x^2}{2}\ - \dfrac{1}{2}\int\ x\, dx\)

                     \(= ln x\, \dfrac{x^2}{2}\ - \dfrac{1}{2}\, \dfrac{x^2}{2}\)

                    \(= ln x\, \dfrac{x^2}{2}\ - \dfrac{x^2}{4}\)

    Simplifying the answer,

    \( \int x \,lnx \, dx\, = \dfrac{x^2}{2}\, (ln x\, -\, \dfrac{1}{2}\, ) +\, C\)

    Reminder: Make sure to write + C at the end of the answer as C is a constant. 

    Example \(\PageIndex{2}\):

    Find \( \int x \,lnx \, dx \).

    Answer:

    Using the priority listed above, let \(u=ln x\) and \(dv=x\,dx\).

    Since \(u=ln x\)  ,      \(du=\dfrac{1}{x} \) .

    Since   \(dv=x\),        \(v= \dfrac{x^2}{2}\).

    Using the IBP, 

    \(\int\,u \,dv= u\,v-\int\, v\, du\)

                   \(= ln x\, \dfrac{x^2}{2}\ -\int\, \dfrac{x^2}{2}\, x\, dx\)

                    \(= ln x\, \dfrac{x^2}{2}\ - \dfrac{1}{2}\int\ x\, dx\)

                     \(= ln x\, \dfrac{x^2}{2}\ - \dfrac{1}{2}\, \dfrac{x^2}{2}\)

                    \(= ln x\, \dfrac{x^2}{2}\ - \dfrac{x^2}{4}\)

    Simplifying the answer,

    \( \int x \,lnx \, dx\, = \dfrac{x^2}{2}\, (ln x\, -\, \dfrac{1}{2}\, ) +\, C\)

    Reminder: Make sure to write + C at the end of the answer as C is a constant. 

    Example \(\PageIndex{3}\):

    Find \( \int \,  lnx \, dx \).

    Answer:

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    Exercise \(\PageIndex{1}\)

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