CHAPTER 64 INTEGRATION USING ALGEBRAIC SUBSTITUTIONS

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1 CHAPTER INTEGRATION USING ALGEBRAIC SUBSTITUTIONS EXERCISE 8 Page 7. Integrate with respet to x: sin( + 9) Let + 9 then d x and dx sin(+ 9) d x sin os + os ( ) os( x+ 9) +. Integrate with respet to θ: os(θ ) Let θ then dθ and dθ + sin( ) os(θ ) dθ os ( sin ) θ +. Integrate with respet to t: se (t + ) Let t then dt and dt t+ t + tan(t ) se ( ) d se ( tan ) + +. Integrate with respet to x: (x ) Let x then d x and dx ( ) + 7 x 7, John Bird

2 . Integrate with respet to x: (x ) Let x then d x and dx d x ln + (x ) ln( x ) +. Integrate with respet to θ: e θ+ Let θ + then dθ and dθ e θ + dθ e e e+ e θ Evalate orret to signifiant figres: ( x+ ) dx Let x + then d x and dx ( ) + x+ + 8 Ths, ( x+ ) dx ( x+ ) [ ] Evalate orret to signifiant figres: ( + ) x x dx Let x + then d x i.e. d x d x x + x x x x ( )d d ( ) ( ). Ths, x ( x + ) dx ( x + ) [ 7 ], John Bird

3 9. Evalate orret to signifiant figres: / sin t+ dt / sin t d t os t os os (note angles are in radians) [.77.77].98. Evalate orret to signifiant figres: ( x ) os d x os( ) d x sin( ) ( sin( ) ) ( sin( ) ) ( sin sin( ) ) (.87.).79.. The mean time to failre, M years, for a set of omponents is given by: M ( t ) Determine the mean time to failre.. d t Let.t then. dt and dt.. (. t). dt (.)(.).. mean time to failre, M ( ) (. t) +..t d t (. t)., John Bird

4 ( ( ) ) ( ) ().. years, John Bird

5 EXERCISE 9 Page 7. Integrate with respet to x: x (x ) Let x then d x i.e. d x x x dx x + + ( x ) ( ) +. Integrate with respet to t: os t sin t Let os t then sin t d t and dt sin t sin t os t os sin d sin t t t t + +. Integrate with respet to x: se x tan x Let tan x then se x d x i.e. d x se x se x tan x+ se x tan xd x se x ( ) d + Alternatively, let se x then sextan x d x i.e. d x sextan x se x tan xd x tan x sextan x sex + se x+. Integrate with respet to t: t ( t ) Let t then t dt and dt t 7, John Bird

6 ( ) ( t ) t t dt t t Integrate with respet to θ: lnθ θ Let ln θ then and dθ θ dθ θ lnθ dθ θd + θ θ ( ) ( ) lnθ +. Integrate with respet to t: tan t sin t tan td t d t os t Let os t then sin t dt i.e. d d t sin t sin t sin t d t ln os t + sin t ln os t+ ln os t + ( ) ln(se t) + 7. Integrate with respet to t: e t (e t + ) Let et + then et d t and d t e t et et dt et ( et ) e + + ( ) 8. Evalate orret to signifiant figres: e ( x x ) d x 8, John Bird

7 Let x then d x i.e. d x ( ) x ( e d e ed e x e ) x x x + + ( x ) ( ) x xe dx e [ e e ].7 9. Evalate orret to signifiant figres: / sin os d θ θ θ Let sin θ then osθ dθ and dθ osθ osθ sin sin θosθdθ osθ + θ + / /. sinθosθdθ sinθ sin ( sin ) ( ) ( ). Evalate orret to signifiant figres: x ( ) d x Let then 8x d x i.e. d x 8x Ths, x x d d x ( ) 8x 8 8 ( ) x d x ( ).99 ( ) ( ). The eletrostati potential on all parts of a onting irlar dis of radis r is given by the 9 R eqation: V σ dr R + r Solve the eqation by determining the integral. 9, John Bird

8 Let R + r then R d R i.e. d R R R r R R dr + R + + R + r ( ) + R 9 9 V σ d σ ( ) σ ( 9 ) R + r R R + r + r r { } ( ) σ 9 + r r. In the stdy of a rigid rotor the following integration ors: r ( + ) Determine Z r for onstant temperatre T assming h, I and k are onstants. J( J+ ) h 8 IkT Z J e dj Let J(J + ) J + J then J d J + i.e. d J J + J( J+ ) h h h h + e d ( + )e e d e + (J + ) h 8 IkT 8 IkT 8 IkT 8 IkT 8 J J J IkT ( ) 8 IkT h J( J+ ) h 8 e IkT + ( ) ( ) J J+ h J J+ h Ths, Z ( e 8 IkT 8 ) d e IkT r J J [ e + e] 8 IkT 8 IkT h h 8 IkT h [ ] 8 IkT h a σ sinθ. In eletrostatis, E dθ where a, σ and ε are onstants, x is ε ( a x ax osθ) greater than a, and x is independent of θ. Show that E a σ ε x Let a + x ax osθ then axsinθ dθ and dθ axsinθ, John Bird

9 a σ sinθ d d a σ a σ a σ E ε axsinθ ε + ax ε ax ε ax ε x ( osθ ) a + x ax + E a x ax osθ a x ax os a x ax os ε x ε x ε x ( a + x + ax) ( a + x ax) ε x ( x+ a) ( x a) [( x+ a) ( x a) ] [ a] ε x ε x i.e. E a σ ε x. The time taken, t hors, for a vehile to reah a veloity of km/h with an initial speed of d v km/h is given by: t where v is the veloity in km/h. Determine t, orret to v the nearest seond. Let v d v and d v dv ln + ln( v) + v Ths, t ln( v) ln ( 9) ln ( 8) dv v ln ( ) ln ( 7 ).97 hors.97h.97 mintes.8 min.8 min.8 seonds. s, John Bird

10 t d v v min s, orret to the nearest seond, John Bird