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.
For $f(x)$ one should take a factor that simplifies the integral when taking the derivative.
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Method
 Determine $f(x)$ and $g'(x)$
 Calculate $f'(x)$: take derivative of $f(x)$
 Calculate $g(x)$: integrate $g'(x)$
 Insert and solve integral
Example
Solve the integral $\int x\cdot\cos(x) \, \mathrm{d}x$ with integration by parts.

Determine $f(x)$ and $g'(x)$
$f(x)=x$
$g'(x)=\cos(x)$
Reason: derivative of $x$ is 1, which simplifies the integral. Calculate $f'(x)$: take derivative of $f(x)$
$f(x)=x$
$f'(x)=\color{blue}{1}$
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. 
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.