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	Prerequisite relationships
Equations C	- Solve quadratic equations by factorisation → Decompose into partial fractions - Solve quadratics by completing the square → Integrate by completing the square
Polynomials	- Divide a polynomial by a linear polynomial → Decompose into partial fractions - Divide a polynomial by a linear polynomial → Integrate improper rational functions
Trigonometric Identities and Equations	- Solve trig equations on a restricted domain → Solve trig equations with product-to-sum - Solve trig equations on a restricted domain → Solve trig equations with the t-formulas
Introduction to Differentiation	- Apply the product rule → Derive integration by parts
Integral Calculus	- Use indefinite integral notation → Derive integration by parts - Apply the reverse chain rule → Integrate by substitution - Integrate $\frac{1}{x}$ and $\frac{1}{ax+b}$ → Integrate using partial fractions - Integrate $\frac{f^{\prime }(x)}{f(x)}$ → Integrate by algebraic manipulation - Integrate trig functions of ax+b → Integrate products of sine and cosine
Further Trigonometry	- Sum and difference identities → Derive product-to-sum identities - Double angle identities → Derive the t-formulas
Further Calculus Skills	- Integrate to inverse trig forms → Integrate by completing the square - Integrate by substitution (revisited content)
The Nature of Proof	- Define a recurrence relation → Recurrence relations with integration by parts

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