Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.3 EXERCISES

2.26 Use the mean value theorem to establish bounds in the following cases.

(a) For−ln(1−y), by considering lnxin the range 0< 1 −y<x<1.

(b) Forey−1, by consideringex−1 in the range 0<x<y.

2.27 For the functiony(x)=x^2 exp(−x) obtain a simple relationship betweenyand
dy/dxand then, by applying Leibnitz’ theorem, prove that

xy(n+1)+(n+x−2)y(n)+ny(n−1)=0.

2.28 Use Rolle’s theorem to deduce that, if the equationf(x)=0hasarepeatedroot
x 1 ,thenx 1 is also a root of the equationf′(x)=0.
(a) Apply this result to the ‘standard’ quadratic equationax^2 +bx+c=0,to
show that a necessary condition for equal roots isb^2 =4ac.
(b) Find all the roots off(x)=x^3 +4x^2 − 3 x−18=0,giventhatoneofthem
is a repeated root.
(c) The equationf(x)=x^4 +4x^3 +7x^2 +6x+ 2 = 0 has a repeated integer root.
How many real roots does it have altogether?

2.29 Show that the curvex^3 +y^3 − 12 x− 8 y−16 = 0 touches thex-axis.
2.30 Find the following indefinite integrals:

(a)

∫

(4 +x^2 )−^1 dx;(b)

∫

(8+2x−x^2 )−^1 /^2 dx for 2≤x≤4;

(c)

∫

(1+sinθ)−^1 dθ;(d)

∫

(x

√

1 −x)−^1 dx for 0<x≤1.

2.31 Find the indefinite integralsJof the following ratios of polynomials:

(a) (x+3)/(x^2 +x−2);

(b) (x^3 +5x^2 +8x+ 12)/(2x^2 +10x+ 12);

(c) (3x^2 +20x+ 28)/(x^2 +6x+9);

(d)x^3 /(a^8 +x^8 ).

2.32 Expressx^2 (ax+b)−^1 as the sum of powers ofxand another integrable term, and
hence evaluate
∫b/a

0

x^2

ax+b

dx.

2.33 Find the integralJof (ax^2 +bx+c)−^1 ,witha= 0, distinguishing between the
cases (i)b^2 > 4 ac, (ii)b^2 < 4 acand (iii)b^2 =4ac.
2.34 Use logarithmic integration to find the indefinite integralsJof the following:

(a) sin 2x/(1 + 4 sin^2 x);

(b)ex/(ex−e−x);

(c) (1 +xlnx)/(xlnx);

(d) [x(xn+an)]−^1.

2.35 Find the derivative off(x) = (1+sinx)/cosxand hence determine the indefinite
integralJof secx.
2.36 Find the indefinite integrals,J, of the following functions involving sinusoids:

(a) cos^5 x−cos^3 x;

(b) (1−cosx)/(1 + cosx);

(c) cosxsinx/(1 + cosx);

(d) sec^2 x/(1−tan^2 x).

2.37 By making the substitutionx=acos^2 θ+bsin^2 θ, evaluate the definite integrals
Jbetween limitsaandb(>a) of the following functions:
(a) [(x−a)(b−x)]−^1 /^2 ;
(b) [(x−a)(b−x)]^1 /^2 ;