Integration by parts: the formula and when to use it

The formula

$\int_a^b u'(x) v(x) \, dx = \Big[ u(x) v(x) \Big]_a^b - \int_a^b u(x) v'(x) \, dx$

Version without bounds (antiderivative):

$\int u' v \, dx = uv - \int u v' \, dx$

When to use it?

When the integrand is a product of two functions where one differentiates nicely (polynomial → constant) and the other integrates nicely ($e^x, \sin, \cos, \ln$).

The “LIATE” choice

To pick $v$ (the one we differentiate): priority to the first appearing in LIATE.

• Logarithm
• Inverse trig ($\arctan$, …)
• Algebraic (polynomial)
• Trigonometric ($\sin$, $\cos$)
• Exponential

The rest becomes $u'$ (we integrate it).

Example 1: $\int x e^x \, dx$

$v = x$ (polynomial), $u' = e^x$.

So $v' = 1$, $u = e^x$.

$\int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C = (x - 1) e^x + C$

Example 2: $\int \ln(x) \, dx$

Trick: write $\ln(x) = 1 \times \ln(x)$.

$v = \ln(x)$ (logarithm), $u' = 1$.

$v' = 1/x$, $u = x$.

$\int \ln(x) \, dx = x \ln(x) - \int x \cdot \dfrac{1}{x} \, dx = x \ln(x) - x + C$

Example 3: $\int x \cos(x) \, dx$

$v = x$, $u' = \cos(x)$.

$v' = 1$, $u = \sin(x)$.

$\int x \cos(x) \, dx = x \sin(x) - \int \sin(x) \, dx = x \sin(x) + \cos(x) + C$

Two successive IBPs ($\int x^2 e^x$)

Sometimes you must redo IBP:

$\int x^2 e^x \, dx = x^2 e^x - 2 \int x e^x \, dx = x^2 e^x - 2(x - 1) e^x + C$

Common mistakes

1. Sign error in the formula.
2. Choosing $v = e^x$ in $\int x e^x$ (makes the integral harder).
3. Forgetting the bounds in $[uv]_a^b$.
4. Forgetting the constant $C$ for an antiderivative.

?À toi de jouer

For ∫ x·ln(x) dx, we set:

A.v = ln(x), u' = xB.v = x, u' = ln(x)C.v = x·ln(x), u' = 1D.IBP not needed

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