Integration
Chapter 80 of 35 strong
RESTS ONSubstitute known values into a formula and evaluate it correctlyCH 4Solve two-step linear equations by inverse operations in reverse orderCH 1Signed numbersCH 3Differentiate e^x, e^(kx) and e^(g(x)) with the factor of the inner derivative, and differentiate sums, expanded squares and quotient forms after rewriting them as sums of e^(kx) termsCH 5Recall and apply the exact derivative of sine, cosine and tangent components, including constant multiples and linear input arguments, as a component in a larger differentiation taskCH 6Identify the inner and outer functions of a composite, differentiate each, multiply the outer derivative evaluated at the inner function by the inner derivative, and write the result in the requested factorised formCH 6Rewrite reciprocal, root and rational terms as a constant times x^n, then differentiate term-by-term with the power rule n a x^(n−1), using linearity for sums and differences and returning the result with positive or fractional indices as requiredCH 5Interpret anti-differentiation as reversing differentiation and distinguish one primitive from the whole vertically translated familyCH 7Add or subtract polynomials by aligning equal powers and combining their coefficients, including zero coefficients for missing powersCH 1Apply the distributive law repeatedly, preserve signs and collect like powers to write a polynomial in descending-power formCH 2Factor numerators and denominators, cancel common factors rather than terms, retain the original excluded values, and present the remaining numerator and denominator in a consistent formCH 3Reduce an angle to its first-quadrant reference angle measured from the horizontal axisCH 1Determine which of sine, cosine and tangent are positive, negative, zero or undefined in each quadrant or on a boundaryCH 1Differentiate a constant multiple, sum or difference term by term while preserving each exact rational coefficient and signFOUNDATIONSDifferentiate a monomial with non-negative integer exponent by multiplying by the exponent and reducing that exponent by one, with a constant term differentiating to zeroCH 6Interpret the derivative as the limit of a simplified difference quotient as the increment approaches zero, cancelling the common increment only after it has been factored and never treating the original quotient as defined at zeroFOUNDATIONSSelect and apply the area formula matching the named shape and measurementsCH 5Completely factorise a low-degree polynomial by taking out common factors, using familiar quadratic patterns, or finding one linear factor and dividing to obtain a lower-degree factorCH 1Solve equations with unknowns on both sides by collecting them onto one sideCH 1Determine both end directions of a polynomial from degree parity and the sign of its leading coefficientCH 2Use the parity and size of a root's multiplicity to determine whether the graph crosses or touches the horizontal axis, distinguish an ordinary crossing from a stationary crossing, and decide whether the sign changesFOUNDATIONSSolve a one-step linear equationCH 7Form a quantity function from an exact polynomial rate, apply an initial or boundary value, respect the modelling domain and evaluate or solve for a stated eventCH 7Substitute an exact value into a derivative and compute the exact rational gradient, distinguishing f'(a) from f(a) and from the unsimplified derivative ruleCH 5Calculate the average rate of change of a function over a non-degenerate interval as the exact change in output divided by the exact change in input, from a rule, table, two points or a factual context, keeping the subtraction order matched between the output change and the input changeCH 6Apply the unitary method to a rateCH 7
Rectangle and trapezium rules
Definite integral evaluation and properties
Fundamental theorem of calculus
Areas under and between curves
Average value, total change and kinematics