Let , where and are real constants. One antiderivative of is . The curve passes…

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let

f(x)=a \cos x + b \sin x

, where

a

and

b

are real constants. One antiderivative of

f

F(x)

. The curve

y=F(x)

passes through the three points

(0,2),\,(\tfrac{\pi}{2},5)

and

(\pi,4)

. (i) Find the general expression for

F(x)

, writing the constant of integration as

C

. (ii) Using each of the three given points successively, establish three equations in

a,\,b

and

C

. (iii) Solve the simultaneous equations to obtain the values of

a,\,b

and

C