Integration by parts is a technique used to solve integrals of the form ∫u(x)v'(x)dx, where u(x) and v(x) are two functions. It's based on the product rule of differentiation:
The Formula:
∫u(x)v'(x)dx = u(x)v(x) - ∫v(x)u'(x)dx
How it works:
- Identify u(x) and v'(x): Choose one function as u(x) and the other as v'(x).
- Find u'(x) and v(x): Differentiate u(x) to find u'(x) and integrate v'(x) to find v(x).
- Apply the formula: Substitute the values of u(x), v(x), u'(x), and v'(x) into the integration by parts formula.
- Solve the new integral: The new integral on the right side of the formula might be easier to solve than the original integral.
Choosing u(x) and v'(x):
The ILATE rule is a helpful mnemonic for choosing u(x):
- Inverse trigonometric functions
- Logarithmic functions
- Algebraic functions
- Trigonometric functions
- Exponential functions
The function higher up on the ILATE list should be chosen as u(x).
Examples:
-
*∫xsin(x)dx:**
- Let u(x) = x and v'(x) = sin(x).
- Then u'(x) = 1 and v(x) = -cos(x).
- Applying the formula: ∫xsin(x)dx = x(-cos(x)) - ∫(-cos(x))1dx = -x*cos(x) + sin(x) + C.
-
∫ln(x)dx:
- Let u(x) = ln(x) and v'(x) = 1.
- Then u'(x) = 1/x and v(x) = x.
- Applying the formula: ∫ln(x)dx = ln(x)x - ∫x(1/x)dx = x*ln(x) - x + C.
Practical insights:
- Integration by parts can sometimes be applied multiple times to solve an integral.
- It is particularly useful for integrals involving products of functions that have different differentiation and integration properties.
