Difficulty: Advanced | Prerequisites: Ch. 5 review, u-substitution, integration by parts, trig identities, polynomial long division.
Trig substitution handles integrals with square roots of quadratics by converting them to trig integrals you already know. Partial fractions breaks rational functions into simpler pieces you can integrate term by term. Numerical integration (trapezoidal rule, Simpson's rule) gives you approximate answers when an antiderivative is impossible or impractical to find. Together these three topics round out the Calc II integration toolkit.
Trigonometric substitution
A technique for integrals containing sqrt(a^2 - x^2), sqrt(a^2 + x^2), or sqrt(x^2 - a^2). You replace x with a trig expression so the square root simplifies via a Pythagorean identity.
Think of it as: dressing x up as a trig function so the radical disappears.
Partial fraction decomposition (PFD)
A method for breaking a rational function P(x)/Q(x) into a sum of simpler fractions whose denominators are the factors of Q(x). Each piece can then be integrated individually.
In simple terms, this means: split one complicated fraction into several easy ones.
Proper rational function
A rational function where the degree of the numerator is strictly less than the degree of the denominator. PFD applies directly to proper fractions.
Improper rational function
A rational function where the numerator's degree is greater than or equal to the denominator's. You must perform polynomial long division first to extract a polynomial plus a proper remainder.
Irreducible quadratic factor
A quadratic ax^2 + bx + c in the denominator that cannot be factored into real linear factors (its discriminant b^2 - 4ac < 0). Its partial fraction term has the form (Ax + B) / (ax^2 + bx + c).
Trapezoidal rule
Approximates the integral by connecting function values with straight lines (trapezoids): T_n = (delta-x / 2)[f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)].
Think of it as: averaging the left and right Riemann sums.
Simpson's rule
Approximates the integral using parabolic arcs through consecutive triplets of points. Requires n to be even: S_n = (delta-x / 3)[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + f(x_n)].
In simple terms, this means: fit little parabolas instead of straight lines, which hugs the curve much more closely.
Error bounds
Formulas that guarantee how far an approximation can be from the true integral. For the midpoint and trapezoidal rules the error depends on the second derivative; for Simpson's rule it depends on the fourth derivative.
The technique applies whenever the integrand contains one of three radical forms. The idea: replace x with a trig expression that turns the radical into a trig function via a Pythagorean identity.
Expression | Substitution | Identity used | dx becomes |
|---|---|---|---|
sqrt(a^2 - x^2) | x = a sin theta | 1 - sin^2 theta = cos^2 theta | a cos theta d-theta |
sqrt(a^2 + x^2) | x = a tan theta | 1 + tan^2 theta = sec^2 theta | a sec^2 theta d-theta |
sqrt(x^2 - a^2) | x = a sec theta | sec^2 theta - 1 = tan^2 theta | a sec theta tan theta d-theta |
Identify which radical form is present.
Make the appropriate substitution and compute dx.
Simplify the radical using the identity.
Evaluate the resulting trig integral (using techniques from Section 7.2).
Convert back to x using a right-triangle diagram.
When the integrand contains a quadratic under a radical that is not in one of the three standard forms, complete the square first. For example, integral of dt/sqrt(4t - 3 - t^2) becomes integral of dt/sqrt(1 - (t-2)^2) after completing the square, which fits the first case with a = 1 and the substitution t - 2 = sin theta.
Always find dx as a function of theta before substituting.
Drawing a reference right triangle labelled with x, a, and the radical helps convert back to x.
Some trig-sub integrals can also be done with ordinary u-sub (e.g. integral of t dt/sqrt(t^2 + 9) yields to u = t^2 + 9 directly). If a simpler method works, use it.
A rational function P(x)/Q(x) is proper when deg(P) < deg(Q). If it is improper (deg(P) >= deg(Q)), perform polynomial long division first to get a polynomial plus a proper remainder.
Case I: Distinct linear factors
Each factor (ax + b) in the denominator contributes a term A/(ax + b). Set up the equation, clear denominators, and solve for the constants by substituting convenient x-values.
Case II: Repeated linear factors
A factor (ax + b)^k contributes k terms: A_1/(ax + b) + A_2/(ax + b)^2 + ... + A_k/(ax + b)^k. You need a separate constant for each power up to k.
Case III: Irreducible quadratic factors
A factor (ax^2 + bx + c) that does not factor over the reals contributes (Ax + B)/(ax^2 + bx + c). The numerator is linear, not just a constant.
Case IV: Repeated irreducible quadratic factors
A factor (ax^2 + bx + c)^k contributes k terms, each with a linear numerator: (A_1 x + B_1)/(ax^2 + bx + c) + ... + (A_k x + B_k)/(ax^2 + bx + c)^k.
After decomposition, each piece integrates with elementary techniques:
A/(x - r) integrates to A ln|x - r| + C.
A/(x - r)^n (n > 1) integrates via the power rule.
(Ax + B)/(x^2 + bx + c) may need to be split into a ln term (from the Ax part, via u-sub) plus an arctan term (from the B part, after completing the square).
Some integrals with roots (e.g. integral of sqrt(x + 9)/x dx) become rational functions after a substitution like u = sqrt(x + 9). You then apply long division and partial fractions to the resulting rational integrand.
Some integrands have no elementary antiderivative (e.g. e^(-x^2), the Gaussian integral). Others are known only from sampled data points. Numerical methods give usable answers in both cases.
All use delta-x = (b - a)/n and x_i = a + i times delta-x.
Left Riemann sum (L_n): Sum of f(x_{i-1}) times delta-x. Uses the left endpoint of each subinterval.
Right Riemann sum (R_n): Sum of f(x_i) times delta-x. Uses the right endpoint.
Midpoint rule (M_n): Sum of f(x-bar_i) times delta-x, where x-bar_i = (x_{i-1} + x_i)/2.
Trapezoidal rule (T_n): (delta-x / 2)[f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]. Equivalent to averaging L_n and R_n.
Simpson's rule (S_n): (delta-x / 3)[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 4f(x_{n-1}) + f(x_n)]. Requires n even. Also expressible as S_{2n} = (1/3) T_n + (2/3) M_n.
Condition | Underestimate | Overestimate |
|---|---|---|
f increasing | L_n | R_n |
f decreasing | R_n | L_n |
f concave up | M_n | T_n |
f concave down | T_n | M_n |
The midpoint rule is generally more accurate than the trapezoidal rule (roughly twice as accurate for smooth functions).
Suppose |f''(x)| <= K on [a, b] and |f^(4)(x)| <= M on [a, b]. Then:
Midpoint error: |E_M| <= K(b - a)^3 / (24n^2)
Trapezoidal error: |E_T| <= K(b - a)^3 / (12n^2)
Simpson's error: |E_S| <= M(b - a)^5 / (180n^4)
Simpson's rule converges much faster (error drops as 1/n^4 vs 1/n^2), so it is the method of choice when you can choose n.
Left/Right Riemann < Trapezoidal < Midpoint < Simpson's.
Students often skip long division on improper rational functions and try to decompose directly. This gives wrong partial fractions. Always check degrees first.
In trig substitution, forgetting to express dx in terms of d-theta before substituting is a frequent error. The substitution is incomplete without it.
When setting up partial fractions for repeated linear factors, students sometimes write only one term (e.g. B/(x-2)^2) instead of both B/(x-2) and D/(x-2)^2. Every power from 1 up to k needs its own term.
For Simpson's rule, n must be even. Using an odd n produces a wrong answer.
Trig substitution combined with completing the square is a favourite exam question (e.g. integral of dx / (x^2 + x + 1)).
Partial fractions with irreducible quadratics (Case III) almost always appears on exams. Practise the technique of splitting (Ax + B)/(x^2 + c) into a u-sub piece and an arctan piece.
Numerical integration questions typically give a table of values (no formula) and ask you to estimate the integral using T_n, M_n, or S_n.
Error bound problems ask you to find n such that the error is below a given tolerance. Solve the bound inequality for n.
Know when each approximation method overestimates vs underestimates: concavity is the key for trapezoidal vs midpoint, and monotonicity for left vs right.
Fill in the blank: For integral of dx / sqrt(9 - x^2), the substitution is x = ______. (3 sin theta.)
True or False: A rational function with numerator degree 4 and denominator degree 3 can be decomposed into partial fractions immediately. (False. Long division first.)
Fill in the blank: Simpson's rule requires n to be ______. (Even.)
True or False: If f is concave up on [a, b], the trapezoidal rule overestimates the integral. (True.)
Fill in the blank: The partial fraction form for A / [(x+1)(x^2+4)] is ______ + ______ . (A/(x+1) + (Bx + C)/(x^2 + 4).)
Q: Evaluate integral of (x+7) / (x^2 - x - 6) dx using partial fractions.
A: Factor denominator: (x-3)(x+2). Decompose: (x+7)/[(x-3)(x+2)] = A/(x-3) + B/(x+2). Setting x = 3 gives A = 2; setting x = -2 gives B = -1. Integral = 2 ln|x-3| - ln|x+2| + C.
Q: Evaluate integral of dx / [x^2 sqrt(4 - x^2)].
A: Use trig sub x = 2 sin theta, dx = 2 cos theta d-theta. sqrt(4 - x^2) = 2 cos theta. The integral becomes integral of (2 cos theta d-theta) / (4 sin^2 theta times 2 cos theta) = (1/4) integral of csc^2 theta d-theta = -(1/4) cot theta + C. Convert back: cot theta = sqrt(4 - x^2) / x, so result = -sqrt(4 - x^2) / (4x) + C.
Q: Use Simpson's rule with n = 6 to estimate integral from 0 to 6 of sqrt(36 - x^2) dx.
A: delta-x = 1. The x-values are 0, 1, 2, 3, 4, 5, 6 with function values 6, sqrt(35), sqrt(32), sqrt(27), sqrt(20), sqrt(11), 0. Apply the 1-4-2-4-2-4-1 pattern with factor delta-x/3 = 1/3. S_6 is approximately 28.175. (The exact value is 9 pi, approximately 28.274.)
Q: Estimate the total distance from the following velocity data using T_6: t (min) = 0, 5, 10, 15, 20, 25, 30 and v (mph) = 3, 4, 5, 4, 7, 8, 11.
A: delta-t = 5. T_6 = (5/2)[3 + 2(4) + 2(5) + 2(4) + 2(7) + 2(8) + 11] = (5/2)[3 + 8 + 10 + 8 + 14 + 16 + 11] = (5/2)(70) = 175 (in units of mph times minutes; convert to miles by dividing by 60 if needed).
Q: Evaluate integral of (2x^2 - x - 1) / [(x^2 + 1)(x - 2)] dx.
A: Decompose: (Ax + B)/(x^2 + 1) + C/(x - 2). Solving gives A = 1, B = 1, C = 1. Integral = (1/2) ln(x^2 + 1) + arctan x + ln|x - 2| + C.
Trig substitution reuses the trig integral techniques from Section 7.2, so comfort with sin^m cos^n strategies is a prerequisite. Partial fractions appear again in differential equations (Laplace transforms) and in series expansions. Numerical integration underpins computational science and engineering: when you model real-world systems with sampled or simulated data, Simpson's rule (or its adaptive variants) is often the tool that turns data into answers.
Related Terms / Search Tags
trig substitution, trigonometric substitution, completing the square, partial fraction decomposition, PFD, distinct linear factors, repeated linear factors, irreducible quadratic, polynomial long division, improper rational function, rationalising substitution, numerical integration, trapezoidal rule, Simpson's rule, midpoint rule, left Riemann sum, right Riemann sum, error bounds, Calculus II, NYU Calculus for Engineers, Chapter 7.3, Chapter 7.4, Chapter 7.7