Further Integration (MEX-C1)

4 dot points across 4 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How does integration by parts reverse the product rule, and how do we choose the parts to evaluate products such as $x e^x$ or $x \ln x$ ?

Apply integration by parts to evaluate integrals of products, including repeated application and the recovery of the original integral

A focused answer to the HSC Maths Extension 2 dot point on integration by parts. The formula, choosing the parts with LIATE, repeated application, the boomerang case where the original integral reappears, definite integrals and reduction formulae, with verified worked examples.

12 min answer →

How does decomposing a rational function into partial fractions turn an awkward integral into a sum of standard logarithm and arctangent integrals?

Decompose rational functions into partial fractions and use the decomposition to integrate, including linear, repeated and irreducible quadratic factors

A focused answer to the HSC Maths Extension 2 dot point on partial fractions. Decomposing proper rational functions over distinct linear, repeated and irreducible quadratic factors, the cover-up method, and integrating to logs and arctangents, with verified worked examples.

12 min answer →

How do trigonometric and rationalising substitutions transform integrals containing square roots or awkward rational functions into standard forms?

Evaluate integrals using trigonometric substitution and the t = tan(x/2) substitution, including completing the square to reach standard inverse-trigonometric and logarithmic forms

A focused answer to the HSC Maths Extension 2 dot point on substitution integration. Trigonometric substitutions for square roots with a reference triangle, completing the square to standard forms, and the t = tan(x/2) substitution for rational trigonometric integrals, with verified worked examples.

12 min answer →

How do we integrate powers and products of trigonometric functions by choosing between the Pythagorean-identity substitution, the double-angle reduction, and the product-to-sum identities?

Integrate powers and products of trigonometric functions: powers of sin and cos (odd and even), powers of tan and sec, the integral of sec x, and products of sines and cosines via product-to-sum identities

A focused answer to the HSC Maths Extension 2 dot point on trigonometric integrals. The odd and even strategies for powers of sin and cos, powers of tan and sec, the standard integrals of sec x and sec cubed x, and products via product-to-sum identities, with every result verified by differentiation.

14 min answer →