Math ↠ Integration rules ↠ Integration by parts

Integration by parts

Integration by parts is a method of integrating a product. It is the inverse of the product rule of differential calculus and states:

$\int f(x)\cdot g'(x) \, \mathrm{d}x =$ $f(x)\cdot g(x) - \int f'(x)\cdot g(x) \, \mathrm{d}x$
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Hint

Integration by parts is often used when the integral is a product of two functions, one of which is easy to take the derivative and the other is easy to integrate.
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Note

The new integral $\int f'(x)\cdot g(x)$ should not be more difficult than the one before it.
For $f(x)$ one should take a factor that simplifies the integral when taking the derivative.
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Method

  1. Determine $f(x)$ and $g'(x)$
  2. Calculate $f'(x)$: take derivative of $f(x)$
  3. Calculate $g(x)$: integrate $g'(x)$
  4. Insert and solve integral

Example

Solve the integral $\int x\cdot\cos(x) \, \mathrm{d}x$ with integration by parts.

  1. Determine $f(x)$ and $g'(x)$

    $f(x)=x$
    $g'(x)=\cos(x)$
    Reason: derivative of $x$ is 1, which simplifies the integral.

  2. Calculate $f'(x)$: take derivative of $f(x)$

    $f(x)=x$
    $f'(x)=\color{blue}{1}$

  3. Calculate $g(x)$: integrate $g'(x)$

    $g'(x)=\cos(x)$
    $g(x)=\color{green}{\sin(x)}$
    Hint: The antiderivative of $\cos(x)$ and some other elementary functions should be remembered.

  4. Insert and solve integral

    First, $f'(x)$ and $g(x)$ are used:
    $\int f(x)\cdot g'(x) \, \mathrm{d}x$ $=f(x)\cdot \color{green}{g(x)} - \int \color{blue}{f'(x)}\cdot \color{green}{g(x)} \, \mathrm{d}x$

    $\int x\cdot\cos(x) \, \mathrm{d}x$ $=x\cdot\color{green}{\sin(x)} - \int \color{blue}{1}\cdot\color{green}{\sin(x)} \, \mathrm{d}x$
    $=x\cdot\sin(x) - \color{red}{\int \sin(x) \, \mathrm{d}x}$

    Now the integral has to be solved.
    $\color{red}{\int \sin(x)\, \mathrm{d}x}=-\cos(x)$

    Insert solved integral:
    $\int x\cdot\cos(x) \, \mathrm{d}x$ $=x\cdot\sin(x) - (-\cos(x))$ $=x\cdot\sin(x)+\cos(x)\color{purple}{+C}$
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Hint

If it is, like in this case, an indefinite integral, then the constant of integration $\color{purple}{C}$ must be set at the end.