Further Integration

calculus year12 ext2

Further integration | NSW Curriculum Website

• 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