Integration by Parts

The Formula

Integration by parts comes from the product rule:

$\int u \, dv = uv - \int v \, du$

Or in function notation:

$\int f(x)g'(x) \, dx = f(x)g(x) - \int f'(x)g(x) \, dx$

Choosing u and dv: LIATE Rule

Choose $u$ in this order of priority:

1. Logarithmic functions ($\ln x$, $\log x$)
2. Inverse trig functions ($\arcsin$, $\arctan$, etc.)
3. Algebraic functions ($x^2$, $x$, polynomials)
4. Trigonometric functions ($\sin$, $\cos$, $\tan$, etc.)
5. Exponential functions ($e^x$, $2^x$)

The remaining part becomes $dv$.

Steps

1. Identify $u$ and $dv$ using LIATE
2. Compute $du$ by differentiating $u$
3. Compute $v$ by integrating $dv$
4. Apply the formula: $uv - \int v \, du$
5. Evaluate the remaining integral

Special Cases

Tabular Method

For $\int x^n e^x \, dx$ or $\int x^n \sin(x) \, dx$, use tabular integration:

Create columns for derivatives of $u$ and integrals of $dv$, alternating signs $(+, -, +, -, \ldots)$.

Circular Integration

When integration by parts leads back to the original integral, solve algebraically.

Examples

Example 1: Polynomial × Exponential

$\int x e^x \, dx$

$u = x$, $dv = e^x \, dx$ $du = dx$, $v = e^x$

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

Example 2: Polynomial × Trigonometric

$\int x \sin(x) \, dx$

$u = x$, $dv = \sin(x) \, dx$ $du = dx$, $v = -\cos(x)$

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

Example 3: Logarithmic

$\int \ln(x) \, dx$

$u = \ln(x)$, $dv = dx$ $du = \frac{1}{x} dx$, $v = x$

$= x \ln(x) - \int x \cdot \frac{1}{x} \, dx = x \ln(x) - \int 1 \, dx$

$= x \ln(x) - x + C = x(\ln(x) - 1) + C$

Example 4: Inverse Trig

$\int \arctan(x) \, dx$

$u = \arctan(x)$, $dv = dx$ $du = \frac{1}{1+x^2} dx$, $v = x$

$= x \arctan(x) - \int \frac{x}{1+x^2} \, dx = x \arctan(x) - \frac{1}{2}\ln(1+x^2) + C$

Example 5: Repeated Integration by Parts

$\int x^2 e^x \, dx$

First application: $u = x^2$, $dv = e^x \, dx$

$= x^2 e^x - \int 2x e^x \, dx$

Second application on $\int 2x e^x \, dx$:

$= x^2 e^x - 2(xe^x - e^x) + C$

$= x^2 e^x - 2xe^x + 2e^x + C = e^x(x^2 - 2x + 2) + C$

Example 6: Circular (Exponential × Trig)

$\int e^x \sin(x) \, dx$

First: $u = \sin(x)$, $dv = e^x \, dx$

$= e^x \sin(x) - \int e^x \cos(x) \, dx$

Second: $u = \cos(x)$, $dv = e^x \, dx$

$= e^x \sin(x) - \left[e^x \cos(x) + \int e^x \sin(x) \, dx\right]$

Let $I = \int e^x \sin(x) \, dx$:

$I = e^x \sin(x) - e^x \cos(x) - I$

$2I = e^x(\sin(x) - \cos(x))$

$I = \frac{e^x}{2}(\sin(x) - \cos(x)) + C$

Example 7: Definite Integral

$\int_0^1 x e^x \, dx = \left[e^x(x - 1)\right]_0^1$

$= e(1 - 1) - e^0(0 - 1) = 0 - (-1) = 1$