(A Level) Further integration

Learning objectives

Integrate e^(ax + b), 1/(ax + b), sin(ax + b), cos(ax + b), sec²(ax + b) and 1/(x² + a²) as standard results, including the factor 1/a

Use trigonometric identities, especially the double-angle formulae, to integrate expressions such as sin²x, cos²3x and tan²x

Integrate rational functions by splitting them into partial fractions, giving exact answers in terms of logarithms and tan⁻¹ where needed

Recognise integrands of the form k f′(x)/f(x), such as 3x²/(x³ + 1) or cot x, and integrate them to k ln|f(x)|

Integrate by parts, choosing which factor to differentiate, for products such as x e^(2x), x² cos x or x sec²x, and integrate ln x by treating it as 1 × ln x

Use a given substitution to transform and evaluate a definite or indefinite integral, converting dx and the limits into the new variable