Cambridge Additional Mathematics

Integration (Chapter 15) 421 EXERCISE 15C 1 Use the fundamental theorem of calculus to find the area between: a thex-axis and y= ...

422 Integration (Chapter 15) Earlier, we showed that theantiderivativeofx^2 is^13 x^3 , and that any function of the form^13 x^3 ...

Integration (Chapter 15) 423 Example 5 Self Tutor If y=x^4 +2x^3 , find dy dx . Hence find Z (2x^3 +3x^2 )dx. If y=x^4 +2x^3 the ...

424 Integration (Chapter 15) We can check that an integral is correct by differentiating the answer. It should give us the , the ...

cis an arbitrary constant called the or . constant of integration integrating constant Integration (Chapter 15) 425 These rules ...

426 Integration (Chapter 15) We expand the brackets and simplify to a form that can be integrated. Example 7 Self Tutor Find: a ...

Integration (Chapter 15) 427 2 Integrate with respect tox: a 3 sinx¡ 2 b 4 x¡2 cosx c sinx¡2 cosx+ex d x^2 p x¡10 sinx e x(x¡1) ...

428 Integration (Chapter 15) If we are given the second derivative we need to integrate twice to find the function. This creates ...

Integration (Chapter 15) 429 Likewise if n 6 =¡ 1 , d dx μ 1 a(n+1) (ax+b)n+1 ¶ = 1 a(n+1) (n+ 1)(ax+b)n£a, =(ax+b)n ) Z (ax+b)n ...

430 Integration (Chapter 15) Example 12 Self Tutor Integrate with respect tox: a 2 e^2 x¡e¡^3 x b 2 sin(3x) + cos(4x+¼) a Z (2e^ ...

Integration (Chapter 15) 431 8 Suppose f^0 (x)=psin ¡ 1 2 x ¢ wherepis a constant. f(0) = 1 and f(2¼)=0. Findpand hence f(x). 9 ...

432 Integration (Chapter 15) Example 13 Self Tutor Find: a Z 3 1 (x^2 +2)dx b Z ¼ 3 0 sinxdx a Z 3 1 (x^2 +2)dx = · x^3 3 +2x ̧ ...

Integration (Chapter 15) 433 7aUse the identity cos^2 x=^12 +^12 cos(2x) to help evaluate Z¼ 4 0 cos^2 xdx. b Use the identity s ...

434 Integration (Chapter 15) Bernhard Riemann x y ab y = f(x) A= Zb a f(x)dx O AREA FINDER 2 -2 2 4 y 6 x semi-circle O Historic ...

Integration (Chapter 15) 435 y x f(x) = 4 - x 2 O 2 6 Find the values ofbsuch that Zb 0 cosxdx= 1 p 2 , 0 <b<¼. 7 Findyif: ...

436 Integration (Chapter 15) 9 Find Z (2x+3)ndx for all integers n 6 =¡ 1. 10 A function has gradient function 2 p x+ a p x and ...

16 Contents: A The area under a curve B The area between two functions C Kinematics Applications of integration 4037 Cambridge c ...

438 Applications of integration (Chapter 16) Opening problem #endboxedheading The illustrated curves are those of y= sinx and y= ...

Applications of integration (Chapter 16) 439 Example 2 Self Tutor Find the area of the region enclosed by y=x^2 +1, thex-axis,x= ...

440 Applications of integration (Chapter 16) Discovery Zb a f(x)dx and areas Does Zb a f(x)dx always give us an area? What to do ...