Integrals

Integration is the inverse process of differentiation and is also the tool for adding up infinitely many small quantities, so it is used to find areas, volumes, work done and total cost from marginal cost. In this chapter you will learn indefinite integrals as antiderivatives, standard formulas, and the main methods of integration: substitution, partial fractions for rational functions, and integration by parts using the ILATE order. You will also learn special integrals such as those of 1/(x^2 + a^2), 1/(x^2 - a^2) and 1/sqrt(a^2 - x^2), and the useful result that the integral of e^x (f(x) + f'(x)) dx is e^x f(x) + C. The second half of the chapter deals with definite integrals. You will meet the first and second fundamental theorems of calculus, which connect the area function with antiderivatives, and evaluate definite integrals by substitution. Finally, you will use the properties of definite integrals, such as changing x to (limit - x) and the results for odd and even functions, to evaluate integrals that cannot be found directly.

Detailed Worksheet: Integrals Section A - Definitions (10 marks) 1. Define an antiderivative of a function. Write the integral of x^n dx for n not equal to -1. (2 marks) 2. State the second fundamental theorem of integral calculus. (2 marks) 3. State the property of definite integrals that relates the integral of f(x) from 0 to k with the integral of f(k - x) over the same limits. (2 marks) 4. Write the formulas for the integral of 1/(x^2 + k^2) dx and of 1/sqrt(k^2 - x^2) dx. (2 marks) 5. Write the formula for integration by parts and state the ILATE rule for choosing the first function. (2 marks) Section B - Calculations and Applications (15 marks) 6. Evaluate (i) the integral of 2x/(1 + x^2) dx and (ii) the integral of sin^2 x dx. (3 marks) 7. Using partial fractions, evaluate the integral of 1/((x + 1)(x + 2)) dx. (3 marks) 8. Using integration by parts, evaluate (i) the integral of x e^x dx and (ii) the integral of x cos x dx. (3 marks) 9. Evaluate (i) the integral of (4x^3 - 5x^2 + 6x + 9) dx from 1 to 2 and (ii) the integral of cos^2 x dx from 0 to pi/2. (3 marks) 10. Using properties of definite integrals, evaluate the integral of sqrt(sin x)/(sqrt(sin x) + sqrt(cos x)) dx from 0 to pi/2. (3 marks) Section C - Diagrams (10 marks) 11. Sketch y = x^2 from x = 0 to x = 2. Shade the area represented by the integral of x^2 dx from 0 to 2, show it as a limit of a sum of rectangles, and evaluate it. (4 marks) 12. Draw a flowchart for choosing a method of integration: standard form, substitution, partial fractions (rational function), or integration by parts (product of two different types of functions). (3 marks) 13. Sketch the graph of y = sin x from -pi to pi and use it to explain why the integral of sin x dx from -pi to pi is zero (odd function property). (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Evaluate the integral of x sin x/(1 + cos^2 x) dx from 0 to pi. (5 marks) 15. Evaluate the integral of (3x - 2)/((x + 1)^2 (x + 3)) dx using partial fractions. (5 marks) 16. Evaluate (i) the integral of e^x (sin x + cos x) dx and (ii) the integral of 1/(x^2 + 2x + 2) dx. Explain the method used in each. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Add the constant of integration C for indefinite integrals and show each substitution clearly.