Every differentiation rule has a corresponding integration rule. For instance, the Substitution Rule for integration corresponds to the Chain Rule for differentiation. The integration rule that corresponds to the Product Rule for differentiation is called integration by parts.
The Product Rule states that if $f$ and $g$ are differentiable functions, then
$$ \frac{d}{dx}[f(x)g(x)] = f(x)g'(x) + g(x)f'(x) $$
In the notation for indefinite integrals this equation becomes
$$ \int [f(x)g'(x) + g(x)f'(x)]\,dx = f(x)g(x) $$
or
$$ \int f(x)g'(x)\,dx + \int g(x)f'(x)\,dx = f(x)g(x) $$
We can rearrange this equation as
$$ \int f(x)g'(x)\,dx = f(x)g(x) - \int g(x)f'(x)\,dx $$
Formula 1 is called the formula for integration by parts. It is perhaps easier to remember in the following notation. Let $u = f(x)$ and $v = g(x)$. Then the differentials are $du = f'(x) dx$ and $dv = g'(x) dx$, so, by the Substitution Rule, the formula for integration by parts becomes
$$ \int u\,dv = uv - \int v\,du $$
Our aim in using integration by parts is to obtain a simpler integral than the one we started with Also, a mnemonic device which is helpful for selecting $u$ when using integration by parts is the
LIATE principle of precedence for: Logarithmic - Inverse trigonometric - Algebraic - Trigonometric - Exponential
If the integrand has several factors, then we try to choose among them a $u$ which appears as high as possible on the list. For example, in $\int xe^{2x}\,dx$ the integrand is $xe^{2x}$, which is the product of an algebraic function $(x)$ and an exponential function ($e^{2x}$). Since Algebraic appears before Exponential, we choose $u = x$. Sometimes the integration turns out to be similar regardless of the selection of $u$ and $dv$, but it is advisable to refer to LIATE when in doubt.
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Find
$$ \int x\sin(x), dx $$
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Evaluate
$$ \int \ln(x), dx $$
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Find
$$ \int t^2e^t, dt $$
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Evaluate
$$ \int e^x\sin(x), dx $$
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