Understanding Engineering Mathematics

(ii) The denominator factorises so we can use partial fractions ∫ dx x^2 + 3 x+ 2 = ∫ dx (x+ 1 )(x+ 2 ) = ∫ [ 1 (x+ 1 ) − 1 (x+ ...

which enable us to integrate functions of the type sinmxcosnx sinmxsinmx (m =n) cosmxcosnx Such integrals are fundamental to c ...

by the double angle formula for cos 2x. = 1 2 ∫ ( 1 −cos 2x)dx = 1 2 ( x− sin 2x 2 ) +C NB: Be careful with functions such as co ...

So ∫ sinxcosxdx= 1 2 ∫ sin 2xdx =− 1 4 cos 2x+C Refer back to Review Question 9.1.6(iv) for an alternative form of this. 9.2.9 U ...

Solution to review question 9.1.9 Whenever you see something like √ a^2 −x^2 try a sine substitution x=asinθ. (i) For ∫ dx √ 9 − ...

First, note that not all products can be integrated by parts, and that integration by parts can be useful for things that are no ...

but only on this first example will we explicitly write u=x dv dx =sinx du dx = 1 v=−cosx to help you on your way. Note that we ...

Note that we only need the one arbitrary constant. We can now transfer theIto the left-hand side to get 2 I=ex(sinx−cosx)+C so I ...

(ii) There are a number of ways we might tackle sin^3 x, covered in Section 9.2.8, but perhaps the easiest is to use the Pythago ...

(xiii) sin 2xcos 2x is, from the double angle formulae,^12 sin 4x and becomes a standard integral on substitutingu= 4 x(271 ➤ ). ...

Not so obvious is the fact that ifa<b<cthen ∫b a f(x)dx+ ∫c b f(x)dx=F(b)−F(a)+F(c)−F(b)=F(c)−F(a) = ∫c a f(x)dx Remember ...

(iii) When we make a substitution, we can change the limits too – then we don’t have to return to the original variables. In ∫ 1 ...

(v) sin 3x (vi) cosx^3 (vii) e^5 x (viii) 1 x+ 1 (ix) ln 2x (x) √ x+1(xi)ln( 3 x+ 1 ) (xii) tan−^1 x B. Integrate the following ...

(i) ∫ x^2 − 3 x+ 2 x− 2 dx (ii) ∫ lnecosxdx (iii) ∫ cos 2x cosx+sinx dx (iv) ∫ ecos (^2) x lne^2 x e3lnxe−sin (^2) xdx (v) ∫ (co ...

9.3.7 Integrating rational functions ➤➤ 252 265 ➤ A.Integrate the following by partial fractions: (i) 2 x (x− 1 )(x+ 3 ) (ii) x+ ...

9.3.11 Choice of integration methods ➤➤ 253 276 ➤ A.From the following three methods of integration: A standard integral B subst ...

9.4 Applications We will be devoting a lot of Chapter 10 to applications of integration, and you will have ample opportunity to ...

These are called theorthogonality relations for sine and cosine. The limits−π,πon the integrals may in fact be replaced byanyint ...

9.3.3 Addition of integrals (i) 2 3 x^3 + 3 2 1 x^2 (ii) 2 sinx+cosx (iii) sinhx=^12 (ex−ex) (iv) −cos(x+ 3 ) (v) ∑n r= 0 arxr+^ ...

9.3.7 Integrating rational functions A. (i) 1 2 ln|(x− 1 )(x+ 3 )^3 | (ii) ln ∣ ∣ ∣ ∣ (x+ 3 )^2 (x+ 2 ) ∣ ∣ ∣ ∣ (iii) 4 3 ln ∣ ∣ ...