IBDP Math Analysis HL - Advanced Integration Techniques

Front

Integration by Parts: The LIATE Rule

Back

A mnemonic for choosing u in the formula ∫ u dv = uv - ∫ v du. Priority: L (Logarithmic), I (Inverse trig), A (Algebraic), T (Trigonometric), E (Exponential). Choosing u as the function earlier in this list simplifies the integral, typically reducing the power of polynomials or differentiating complex functions.

Front

Deriving the Trapezoidal Rule from Linear Approximation

Back

Approximates the area under f(x) by dividing the region into n trapezoids of width h. Formula: ∫ f(x) dx ≈ (h/2) [y_0 + 2(y_1 + ... + y_{n-1}) + y_n]. It assumes linear behavior between data points; error is proportional to the second derivative f''(x).

Front

Partial Fractions: Non-Repeated Linear Factors

Back

Used to integrate rational functions where the denominator factors into distinct linear terms, e.g., (x+1)(x-2). Decompose as A/(x+1) + B/(x-2). Solving for A and B allows integration of the simpler terms into logarithmic functions (e.g., ln|x+1| + ln|x-2|).

Front

Volumes of Revolution: The Washer Method

Back

Used to find the volume of a solid of revolution when the cross-sections have a hole (an inner radius). The volume is given by V = π∫[a to b](R_outer^2 - R_inner^2) dx, where R_outer and R_inner are the distances from the axis of rotation to the outer and inner curves, respectively.

Front

Integration by Parts: Cyclical Case

Back

When applying ∫ u dv yields an integral containing the original term (e.g., ∫ e^x sin x dx), solve algebraically by moving the repeating term to the Left Hand Side (LHS). Let I = original integral. Rearranging 'I = Expression - I' gives 2I = Expression, so I = Expression / 2 + C.