Exercise practice: Integrals

6.1 How the exercises are arranged

The chapter has eleven exercise sets plus the miscellaneous exercise. Exercise 7.1 covers the standard forms and their checks by differentiation. Exercise 7.2 covers substitution. Exercise 7.3 uses trigonometric identities. Exercise 7.4 covers the particular functions, including the four special integrals. Exercise 7.5 covers partial fractions. Exercise 7.6 covers integration by parts. Exercise 7.7 covers the mixed special forms. Exercise 7.8 covers definite integrals evaluated directly. Exercise 7.9 covers definite integrals by substitution. Exercise 7.10 covers the properties of definite integrals. Exercise 7.11 covers the limit of a sum. The miscellaneous exercise mixes all of these.

6.2 Standard forms and by parts

Exercise 7.1 is a memory check. Integrate x squared plus one, one over x squared, the square root of x, and the standard circular forms; after each answer differentiate it mentally to confirm. Exercise 7.6 asks for the by-parts pairs: for x squared e to the x, take u equal to x squared and apply two rounds; for x sine x, one round suffices; for x log x, take u equal to log x. The LIATE rule decides each choice.

6.3 Substitution and identities

Exercise 7.2 items are recognisable by the presence of a derivative next to the function. For x e to the power x squared the substitution is the exponent; for sec squared x over tan x it is the tangent. Exercise 7.3 items use the double angle identities. The integral of sine squared is the integral of one minus cosine two x over two, and the integral of sine to the fourth is the square of that expression expanded.

6.4 The four special forms

Exercise 7.4 gives integrals of the types one over a quadratic, one over the square root of a quadratic, and the two square root forms. The method is always the same: complete the square, then read off the inverse tangent, inverse sine or logarithm form from the formula sheet. For the integral of one over x squared plus two x plus five, the completion gives (x+1) squared plus four, and the answer is half the inverse tangent of (x+1)/2.

6.5 Partial fractions and definite integrals

Exercise 7.5 items are split by the shape of the denominator. Distinct linear factors get one constant each; a repeated factor gets a constant for each power; an irreducible quadratic gets a linear numerator. Exercise 7.8 and 7.9 are evaluated with the fundamental theorem; in 7.9, change the limits with the substitution. Exercise 7.10 is a properties exercise: use x to a minus x and the even-odd rules before doing any antiderivative work.

6.6 Limit of a sum and miscellaneous

Exercise 7.11 asks for the limit of a sum for simple polynomials and for the exponential. Write h, write the summand, substitute the summation formulae, simplify the rational function of n and take the limit. The miscellaneous exercise mixes a rational function, an inverse trigonometric integrand and a definite integral; the order of attack is standard form first, then substitution, then by parts, then partial fractions.

6.7 Two more worked exercise patterns

A rational function with an irreducible quadratic. Integrate x plus one over x squared plus two x plus three. Write the numerator as half the derivative of the denominator plus a constant: half of two x plus two is x plus one, so the numerator is exactly half the derivative. The integral is half the logarithm of x squared plus two x plus three, and no inverse tangent term is needed because no constant remains. If the constant had remained, that piece would give an inverse tangent after completing the square.

A definite integral with an absolute value. For the integral from zero to three of the absolute value of x minus one, split at one. From zero to one the integrand is one minus x and its integral is one half. From one to three the integrand is x minus one and its integral is two. The total is five halves, which can also be checked as the sum of two triangle areas.

6.8 Common mistakes in the exercise sets

The most common error in Exercise 7.2 is forgetting to convert dx completely; the second is a wrong choice of u. In Exercise 7.5 an incomplete partial fraction split is the usual slip, and in Exercise 7.10 students often substitute the limits before changing the variable. Each of these is caught by the one-line check of differentiating the answer or by re-reading the substitution line.

6.9 A last look at the miscellaneous exercise

The miscellaneous exercise is where the chapter tests whether the methods can be chosen rather than applied on request. Read each integrand and say the method aloud before starting: substitution for a function with its derivative, partial fractions for a rational function, by parts for a product of unrelated types, properties for a definite integral with symmetry. Choosing the method is half the marks, so state it in the answer.