Understanding Engineering Mathematics

8.3.6 Parametric differentiation A.^12 e−t,−^14 e−^3 t B.(i) −cott,− 1 3 cosec^2 t (ii) 1 t ,− 1 2 t^3 (iii) 1 sint+cost , e−t(s ...

9 Techniques of Integration Most of us find integration difficult – basically because we are always trying to ‘undo’ some differ ...

9.1 Review 9.1.1 Definition of integration ➤253 280➤➤ A.Differentiate (i) x^3 (ii) sin 2x (iii) 2e^3 x B. Find the integral of ( ...

9.1.6 Thedu=f′.x/dxsubstitution ➤263 282➤➤ Integrate the following functions by means of an appropriate substitution. (i) x+ 1 x ...

9.1.11 Choice of integration methods ➤276 284➤➤ Discuss the methods you would use to integrate the following – you will be asked ...

For example, because we know that the derivative ofx^2 is 2x, so we know that the integral of 2xisx^2. Actually, because the der ...

fromA(iii), we see that d dx ( 2 3 e^3 x ) = 2 e^3 x So, again remembering the arbitrary constant: ∫ 2 e^3 xdx= 2 3 e^3 x+C 9.2. ...

Ta b l e 9. 1 f(x) ∫ f(x)dx(arbitrary constant omitted) xα α =− 1 xα+^1 /(α+ 1 )α =− 1 1 x ln|x| cosx sinx sinx −cosx sec^ ...

As usual, the places where you might have trouble occur when negative signs and fractions are involved. In such cases, just take ...

Applying the result a number of times we see for example that ∫ 3 f(x)dx= ∫ (f (x)+f(x)+f(x))dx = ∫ f(x)dx+ ∫ f(x)dx+ ∫ f(x)dx = ...

We consider changes of variable,x, later under various substitution methods, but here we will have a first look at what we might ...

9.2.5 Linear substitution in integration ➤ 251 282➤ It may be thatf(x)in ∫ f(x)dxis inconvenient for integration because of the ...

∫ ( 2 x− 4 )^3 dx= 1 2 ∫ ( 2 x− 4 )^3 d( 2 x− 4 ) = 1 2 1 4 ( 2 x− 4 )^4 +C = 1 8 ( 2 x− 4 )^4 +C (d( 2 x− 4 )= 2 dx) There is a ...

Solution to review question 9.1.5 These questions all involve linear substitutions. In no case is it necessary to do anything to ...

9.2.6 Thedu=f′.x/dxsubstitution ➤ 252 282➤ Integration by substitution is often difficult. The substitution to use is not always ...

Solution to review question 9.1.6 (i) In the integral ∫ x+ 1 x^2 + 2 x+ 3 dxthe derivative ofx^2 + 2 x+3is 2 (x+ 1 )and we have ...

Or: ∫ xsin(x^2 + 1 )dx= 1 2 ∫ sin(x^2 + 1 )d(x^2 + 1 ) =− 1 2 cos(x^2 + 1 )+C (iii) Now you may be able to appreciate the follow ...

Example ∫ x x^2 − 3 x+ 2 dx= ∫ x (x− 1 )(x− 2 ) dx ( factorise denominator ) ( 45 ➤ ) = ∫ [ 2 x− 2 − 1 x− 1 ] dx ( split into pa ...

Solution to review question 9.1.7 If, in these questions, you have done something like: ‘ ∫ dx x^2 + 2 x+ 2 = 1 2 x+ 2 ln(x^2 + ...

C.This question illustrates that sometimes you may need to combine methods. Here, the numerator is of the same degree as the den ...