Glossary
Integration by Parts
A powerful technique used to integrate the product of two functions, derived from the product rule for differentiation. It transforms a complex integral into a potentially simpler one using the formula ∫ u dv = uv - ∫ v du.
Example:
To evaluate ∫ x cos(x) dx, one would use Integration by Parts by setting u=x and dv=cos(x)dx.
LIATE Mnemonic
An acronym (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) used as a guideline to help choose the 'u' term in Integration by Parts. It prioritizes functions that simplify when differentiated.
Example:
When integrating ∫ x ln(x) dx, LIATE suggests choosing u = ln(x) because Logarithmic functions are typically prioritized over Algebraic ones.
Product Rule (for differentiation)
A rule for finding the derivative of a product of two functions, stating that if h(x) = f(x)g(x), then h'(x) = f'(x)g(x) + f(x)g'(x). Integration by Parts is essentially its reverse process.
Example:
The derivative of x sin(x) using the Product Rule is 1 ⋅ sin(x) + x ⋅ cos(x).
Riemann Sums
A method for approximating the definite integral of a function by dividing the area under its curve into a series of rectangles and summing their areas. It forms the basis for the definition of the definite integral.
Example:
Approximating the area under y=x^2 from x=0 to x=2 using four rectangles of equal width is an application of Riemann Sums.
du (in Integration by Parts)
The differential of u, obtained by differentiating the chosen u with respect to the variable of integration. It is used in the ∫ v du part of the Integration by Parts formula.
Example:
If u = ln(x), then du = (1/x) dx, which is used in the new integral.
dv (in Integration by Parts)
The part of the integrand chosen to be integrated in the Integration by Parts formula, including the differential dx. It must be a function that can be readily integrated.
Example:
In ∫ x sin(x) dx, choosing dv = sin(x)dx is effective because its integral, v=-cos(x), is straightforward.
u (in Integration by Parts)
The part of the integrand chosen to be differentiated in the Integration by Parts formula. It should be selected carefully, often using the LIATE mnemonic, to simplify the resulting integral.
Example:
In ∫ x e^x dx, choosing u = x simplifies the integral because its derivative, du=dx, is simpler.
u-substitution
An integration technique that simplifies an integral by replacing a complex expression with a single variable, u, and its differential, du. It is the reverse of the chain rule for differentiation.
Example:
To evaluate ∫ 2x cos(x^2) dx, one could use u-substitution by letting u=x^2, so du=2xdx.
v (in Integration by Parts)
The antiderivative (integral) of dv. It is used in both the uv term and the ∫ v du term of the Integration by Parts formula.
Example:
If dv = e^x dx, then v = e^x, which is then used in the formula.