#Calculus #Year12 #Ext2 >[!info]- [Further integration | NSW Curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-extension-2-11-12-2024/content/n12/fa812ef8f0) >- ME2-12-04 selects and uses techniques of integration to solve problems ## 📖 Prior Knowledge | Content | Prior knowledge | Used for | | -------------------------------------------------- | ---------------------------- | ------------------------------------------------------------- | | [[Further Trigonometry]] | - compound angle formula | - sums and differences of trigonometric functions | | [[Further Calculus Skills]] | - further integration skills | - integration by substitution where substitution is not given | ## Further integration - Derive the identities for trigonometric products as sums and differences for $\cos A \cos B = \frac{1}{2}[\cos(A-B)+\cos(A+B)]$, $\sin A \sin B = \frac{1}{2}[\cos(A-B)-\cos(A+B)]$, $\sin A \cos B = \frac{1}{2}[\sin(A+B)+\sin(A-B)]$ and $\cos A \sin B = \frac{1}{2}[\sin(A+B)-\sin(A-B)]$ - Use the identities for trigonometric products as sums and differences to solve problems and prove results - Solve trigonometric equations by applying the formulas for trigonometric products as sums and differences for $\cos A \cos B$, $\sin A \sin B$ and $\sin A \cos B$ over restricted domains - Use identities relating the trigonometric products as sums and differences to solve problems involving integrals of the form $\int \sin(mx) \cos(nx) dx$, $\int \sin(mx) \sin(nx) dx$ or $\int \cos(mx) \cos(nx) dx$ - Derive the expressions $\sin A = \frac{2t}{1+t^2}$, $\cos A = \frac{1-t^2}{1+t^2}$ and $\tan A = \frac{2t}{1-t^2}$ where $t = \tan \frac{A}{2}$ (the t-formulas) and use them to solve trigonometric equations over restricted domains - Find indefinite integrals and evaluate definite integrals using the method of integration by substitution, where the substitution may or may not be given - Decompose rational functions whose denominators can be expressed as a product of distinct linear factors, distinct irreducible quadratic factors and perfect square factors into partial fractions - Integrate rational functions whose denominators can be expressed as a product of distinct linear factors, distinct irreducible quadratic factors and perfect square factors, using partial fraction decomposition - Integrate rational functions by completing the square on a quadratic denominator - Integrate rational functions where the degree of the numerator is not less than the degree of the denominator - Integrate functions by changing an integrand into an appropriate form using algebraic manipulation - Derive the method for integration by parts - Find indefinite integrals and evaluate definite integrals using the method of integration by parts, including problems where more than one application is required - Derive and use recurrence relations involving integration by parts - Solve theoretical problems involving multiple techniques of integration - Solve practical problems involving multiple techniques of integration