7.1. INDEFINITE INTEGRAL 283 are evaluated using the formulas sin α cos β = 1 2 [sin(α + β)+sin(α – β)], cos α cos β = 1 2 [cos(α + β)+cos(α – β)], sin α sin β = 1 2 [cos(α – β)–cos(α + β)]. 4. Integrals of the form  sin m x cos n xdx,wherem and n are integers, are evaluated as follows: (a) if m is odd, one uses the change of variable cos x = z, with sin xdx =–dz; (b) if n is odd, one uses the change of variable sin x = z, with cos xdx = dz; (c) if m and n are both even nonnegative integers, one should use the degree reduction formulas sin 2 x = 1 2 (1 –cos2x), cos 2 x = 1 2 (1 +cos2x), sin x cos x = 1 2 sin 2x. Example 3. Evaluate the integral  sin 5 xdx. This integral corresponds to odd m: m = 5. With simple rearrangement and the change of variable cos x = z,wehave  sin 5 xdx=  (sin 2 x) 2 sin xdx=–  (1 –cos 2 x) 2 d cos x =–  (1 – z 2 ) 2 dz = 2 3 z 3 – 1 5 z 5 – z + C = 2 3 cos 3 x – 1 5 cos 5 x –cosx + C. Remark. In general, the integrals  sin p x cos q xdx are reduced to the integral of a differential binomial by the substitution y =sinx. 7.1.6. Integration of Polynomials Multiplied by Elementary Functions  Throughout this section, P n (x) designates a polynomial of degree n . 7.1.6-1. Integration of the product of a polynomial by exponential functions. General formulas:  P n (x)e ax dx = e ax  P n (x) a – P  n (x) a 2 + ···+(–1) n P (n) n (x) a n+1  + C,  P n (x)cosh(ax) dx =sinh(ax)  P n (x) a + P  n (x) a 3 + ···  –cosh(ax)  P  n (x) a 2 + P  n (x) a 4 + ···  + C,  P n (x)sinh(ax) dx =cosh(ax)  P n (x) a + P  n (x) a 3 + ···  –sinh(ax)  P  n (x) a 2 + P  n (x) a 4 + ···  + C. These formulas are obtained by repeated integration by parts; see formula 4 from Para- graph 7.1.2-2 with f(x)=P n (x)forg (n+1) (x)=e ax , g (n+1) (x)=cosh(ax), and g (n+1) (x)= sinh(ax), respectively. In the special case P n (x)=x n ,thefirst formula gives  x n e ax dx = e ax n  k=0 (–1) n–k a n+1–k n! k! x k + C. 284 INTEGRALS 7.1.6-2. Integration of the product of a polynomial by a trigonometric function. 1 ◦ . General formulas:  P n (x)cos(ax) dx =sin(ax)  P n (x) a – P  n (x) a 3 + ···  +cos(ax)  P  n (x) a 2 – P  n (x) a 4 + ···  + C,  P n (x)sin(ax) dx =sin(ax)  P  n (x) a 2 – P  n (x) a 4 + ···  –cos(ax)  P n (x) a – P  n (x) a 3 + ···  + C. These formulas are obtained by repeated integration by parts; see formula 4 from Para- graph 7.1.2-2 with f (x)=P n (x)forg (n+1) (x)=cos(ax)andg (n+1) (x)=sin(ax), respectively. 2 ◦ . To evaluate integrals of the form  P n (x)cos m (ax) dx,  P n (x)sin m (ax) dx, with m = 2, 3, , one should first use the trigonometric formulas cos 2k (ax)= 1 2 2k–1 k–1  i=0 C i 2k cos[2(k – i)ax]+ 1 2 2k C k 2k (m = 2k), cos 2k+1 (ax)= 1 2 2k k  i=0 C i 2k+1 cos[(2k – 2i + 1)ax](m = 2k + 1), sin 2k (ax)= 1 2 2k–1 k–1  i=0 (–1) k–i C i 2k cos[2(k – i)ax]+ 1 2 2k C k 2k (m = 2k), sin 2k+1 (ax)= 1 2 2k k  i=0 (–1) k–i C i 2k+1 sin[(2k – 2i + 1)ax](m = 2k + 1), thus reducing the above integrals to those considered in Item 1 ◦ . 3 ◦ . Integrals of the form  P n (x) e ax sin(bx) dx,  P n (x) e ax cos(bx) dx can be evaluated by repeated integration by parts. In particular,  x n e ax sin(bx)=e ax n+1  k=1 (–1) k+1 n! (n – k + 1)! (a 2 + b 2 ) k/2 x n–k+1 sin(bx + kθ)+C,  x n e ax cos(bx)=e ax n+1  k=1 (–1) k+1 n! (n – k + 1)! (a 2 + b 2 ) k/2 x n–k+1 cos(bx + kθ)+C, where sin θ =– b √ a 2 + b 2 ,cosθ = a √ a 2 + b 2 . 7.1. INDEFINITE INTEGRAL 285 7.1.6-3. Integrals involving power and logarithmic functions. 1 ◦ . The formula of integration by parts with g  (x)=P n (x) is effective in the evaluation of integrals of the form  P n (x)ln(ax) dx = Q n+1 (x)ln(ax)–a  Q n+1 (x) x dx, where Q n+1 (x)=  P n (x) dx is a polynomial of degree n+1. The integral on the right-hand side is easy to take, since the integrand is the sum of power functions. Example. Evaluate the integral  ln xdx. Setting f(x)=lnx and g  (x)=1,wefind f  (x)= 1 x and g(x)=x. Substituting these expressions into the formula of integration by parts, we obtain  ln xdx= x ln x –  dx = x ln x – x + C. 2 ◦ . The easiest way to evaluate integrals of the more general form I =  n  i=0 ln i (ax)  m  j=0 b ij x β ij  dx, where the β ij are arbitrary numbers, is to use the substitution z =ln(ax), so that I =  n  i=0 z i  m  j=0 b ij a β ij +1 e (β ij +1)z  dz. By removing the brackets, one obtains a sum of integrals like  x n e ax dx, which are easy to evaluate by the last formula in Paragraph 7.1.6-1. 7.1.6-4. Integrals involving inverse trigonometric functions. 1 ◦ . The formula of integration by parts with g  (x)=P n (x) also allows the evaluation of the following integrals involving inverse trigonometric functions:  P n (x)arcsin(ax) dx = Q n+1 (x)arcsin(ax)–a  Q n+1 (x) √ 1 – a 2 x 2 dx,  P n (x) arccos(ax) dx = Q n+1 (x) arccos(ax)+a  Q n+1 (x) √ 1 – a 2 x 2 dx,  P n (x)arctan(ax) dx = Q n+1 (x)arctan(ax)–a  Q n+1 (x) a 2 x 2 + 1 dx,  P n (x) arccot(ax) dx = Q n+1 (x) arccot(ax)+a  Q n+1 (x) a 2 x 2 + 1 dx, where Q n+1 (x)=  P n (x) dx is a polynomial of degree n+1. The integrals with radicals on the right-hand side in the first two formulas can be evaluated using the techniques described in Paragraph 7.1.4-2. The integrals of rational functions on the right-hand side in the last two formulas can be evaluated using the techniques described in Subsection 7.1.3. 286 INTEGRALS Remark. The above formulas can be generalized to contain any rational functions R  (x)andR(x) instead of the polynomials P n (x)andQ n+1 (x), respectively. 2 ◦ . The following integrals are taken using a change of variable:  P n (x)arcsin m (ax) dx = 1 a  cos zP n  sin z a  z m dz, substitution z =arcsin(ax);  P n (x) arccos m (ax) dx =– 1 a  sin zP n  cos z a  z m dz, substitution z = arccos(ax), where m = 2, 3, The expressions cos z sin k z and sin z cos k z (k = 1, , n)inthe integrals on the right-hand sides should be expressed as sums of sines and cosines with appropriate arguments. Then it remains to evaluate integrals considered in Paragraph 7.1.6-2. 7.2. Definite Integral 7.2.1. Basic Definitions. Classes of Integrable Functions. Geometrical Meaning of the Definite Integral 7.2.1-1. Basic definitions. Let y = f(x) be a bounded function definedonafinite closed interval [a, b]. Let us partition this interval into n elementary subintervals defined by a set of points {x 0 , x 1 , , x n } such that a = x 0 < x 1 < ··· < x n = b. Each subinterval [x k–1 , x k ] will be characterized by its length Δx k = x k – x k–1 and an arbitrarily chosen point ξ k [x k–1 , x k ]. Let us make up an integral sum (a Cauchy–Riemann sum, also known as a Riemann sum) s n = n  k=1 f(ξ k )Δx k (x k–1 ≤ ξ k ≤ x k ). If, as n →∞and, accordingly, Δx k → 0 for all k, there exists a finite limit of the integral sums s n and it depends on neither the way the interval [a, b] was split up, nor the selection of the points ξ k , then this limit is denoted  b a f(x) dx and is called the definite integral (also the Riemann integral) of the function y = f(x) over the interval [a, b]:  b a f(x) dx = lim n→∞ s n  max 1≤k≤n Δx k → 0 as n →∞  . In this case, the function f(x) is called integrable on the interval [a, b]. 7.2.1-2. Classes of integrable functions. 1. If a function f (x) is continuous on an interval [a, b], then it is integrable on this interval. 2. If a bounded function f (x)hasfinitely many jump discontinuities on [a, b], then it is integrable on [a, b]. 3. A monotonic bounded function f(x)isalwaysintegrable. 7.2. DEFINITE INTEGRAL 287 7.2.1-3. Geometric meaning of the definite integral. If f(x) ≥ 0 on [a, b], then the integral  b a f(x) dx is equal to the area of the domain D = {a ≤ x ≤ b, 0 ≤ y ≤ f (x)} (the area of the curvilinear trapezoid shown in Fig. 7.1). D yfx= () y x abO Figure 7.1. The integral of a nonnegative function f (x)onaninterval[a, b] is equal to the area of the shaded region. 7.2.2. Properties of Definite Integrals and Useful Formulas 7.2.2-1. Qualitative properties of integrals. 1. If a function f(x) is integrable on [a, b], then the functions cf(x), with c = const, and |f(x)| are also integrable on [a, b]. 2. If two functions f(x)andg(x) are integrable on [a, b], then their sum, difference, and product are also integrable on [a, b]. 3. If a function f(x) is integrable on [a, b] and its values lie within an interval [c, d], where a function g(y)isdefined and continuous, then the composite function g(f(x)) is also integrable on [a, b]. 4. If a function f (x) is integrable on [a, b], then it is also integrable and on any subin- terval [α, β] ⊂ [a, b]. Conversely, if an interval [a, b] is partitioned into a number of subintervals and f (x) is integrable on each of the subintervals, then it is integrable on the whole interval [a, b]. 5. If the values of a function are changed at finitely many points, this will not affect the integrability of the function and will not change the value of the integral. 7.2.2-2. Properties of integrals in terms of identities. 1. The integral over a zero-length interval is zero:  a a f(x) dx = 0. 2. Antisymmetry under the swap of the integration limits:  b a f(x) dx =–  a b f(x) dx. This property can be taken as the defi nition of a definite integral with a > b. 288 INTEGRALS 3. Linearity. If functions f(x)andg(x) are integrable on an interval [a, b], then  b a  Af(x) Bg(x)  dx = A  b a f(x) dx B  b a g(x) dx for any numbers A and B. 4. Additivity.Ifc [a, b]andf(x) is integrable on [a, b], then  b a f(x) dx =  c a f(x) dx +  b c f(x) dx. 5. Differentiation with respect the variable upper limit.Iff(x) is continuous on [a, b], then the function Φ(x)=  x a f(t) dt is differentiable on [a, b], and Φ  (x)=f(x). This can be written in one relation: d dx   x a f(t) dt  = f (x). 6. Newton–Leibniz formula:  b a f(x) dx = F (x)    b a = F (b)–F (a), where F (x) is an antiderivative of f(x)on[a, b]. 7. Integration by parts. If functions f(x)andg(x) have continuous derivatives on [a, b], then  b a f(x)g  (x) dx =  f(x)g(x)     b a –  b a f  (x)g(x) dx. 8. Repeated integration by parts:  b a f(x)g (n+1) (x) dx =  f(x)g (n) (x)–f  (x)g (n–1) (x)+···+(–1) n f (n) (x)g(x)  b a +(–1) n+1  b a f (n+1) (x)g(x) dx, n = 0, 1, 9. Change of variable (substitution) in a definite integral.Letf(x) be a continuous function on [a, b]andletx(t) be a continuously differentiable function on [α, β]. Suppose also that the range of values of x(t) coincides with [a, b], with x(α)=a and x(β)=b.Then  b a f(x) dx =  β α f  x(t)  x  (t) dt. Example. Evaluate the integral  3 0 dx (x – 8) √ x + 1 . Perform the substitution x + 1 = t 2 , with dx = 2tdt.Wehavet = 1 at x = 0 and t = 2 at x = 3. Therefore  3 0 dx (x – 8) √ x + 1 =  2 1 2tdt (t 2 – 9)t = 2  2 1 dt t 2 – 9 = 1 3 ln    t – 3 t + 3        2 1 = 1 3 ln 2 5 . 10. Differentiation with respect to a parameter.Letf (x, λ) be a continuous function in a domain a ≤ x ≤ b, λ 1 ≤ λ ≤ λ 2 and let it has a continuous partial derivative ∂ ∂λ f(x, λ)inthe 7.2. DEFINITE INTEGRAL 289 same domain. Also let u(λ)andv(λ) be differentiable functions on the interval λ 1 ≤ λ ≤ λ 2 such that a ≤ u(λ) ≤ b and a ≤ v(λ) ≤ b.Then d dλ  u(λ) v(λ) f(x, λ) dx = f  u(λ), λ  du(λ) dλ – f  v(λ), λ  dv(λ) dλ +  u(λ) v(λ) ∂ ∂λ f(x, λ) dx. 11. Cauchy’s formula for multiple integration:  x a dx 1  x 1 a dx 2  x n–1 a f(x n ) dx n = 1 (n – 1)!  x a (x – t) n–1 f(t) dt. 7.2.3. General Reduction Formulas for the Evaluation of Integrals Below are some general formulas, involving arbitrary functions and parameters, that could facilitate the evaluation of integrals. 7.2.3-1. Integrals involving functions of a linear or rational argument.  b a f(a + b – x) dx =  b a f(x) dx;  a 0 [f(x)+f(a – x)] dx = 2  a 0 f(x) dx;  a 0 [f(x)–f(a – x)] dx = 0;  a –a f(x) dx = 0 if f (x) is odd;  a –a f(x) dx = 2  a 0 f(x) dx if f(x)iseven;  b a f(x, a + b – x) dx = 0 if f(x, y)=–f(y, x);  1 0 f  2x √ 1 – x 2  dx =  1 0 f  1 – x 2 ) dx. 7.2.3-2. Integrals involving functions with trigonometric argument.  π 0 f(sin x) dx = 2  π/2 0 f(sin x) dx;  π/2 0 f(sin x) dx =  π/2 0 f(cos x) dx;  π/2 0 f(sin x,cosx) dx = 0 if f (x, y)=–f(y, x);  π/2 0 f(sin 2x)cosxdx =  π/2 0 f(cos 2 x)cosxdx; . Integrals involving power and logarithmic functions. 1 ◦ . The formula of integration by parts with g  (x)=P n (x) is effective in the evaluation of integrals of the form  P n (x)ln(ax) dx =. integral of a nonnegative function f (x)onaninterval[a, b] is equal to the area of the shaded region. 7.2.2. Properties of Definite Integrals and Useful Formulas 7.2.2-1. Qualitative properties of integrals. 1 General Reduction Formulas for the Evaluation of Integrals Below are some general formulas, involving arbitrary functions and parameters, that could facilitate the evaluation of integrals. 7.2.3-1.