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(a) Find the indeﬁnite integrals of the following functions.

(i) f (t) = 6 cos(3t) + 5e^−10t

(ii) g(x) = 21-12x^3/ x (x > 0)

(iii) h(u) = cos^2( 1/8 u)

(b) Evaluate: (this big F sign at the start, 5 at the top and 1 at the bottom)
5 1/4x (7 + 6x^2) dx
F
1

(c)
(i) Write down a deﬁnite integral that will give the value of the area under the curve y = x^2 cos(2x) between x = 3/4 pie and x = pie.

(You are not asked to evaluate the integral by hand.)

(ii) ﬁnd the area described in part (c)(i), giving your
answer correct to three decimal places

for a) ii)

6cos (3t) dx = -4sin (3t) + c
5e^10t dx = 1/5 e^-10t + c

ii) g (x) = 21- 12x^3 / x (x>0)

diverse through by x

g (x) = 21/ x - 12x^3

21/ x - 12x^3 dx = 12Ln x - x^4 + c

iii) h (u) = cos^2 (1/8u)

using the second version of the double angled formula for cos (2delta)

=1/2 cos (2x 1/8 u) + 1/2 du
= 1/2 / 28 sin (2/8u) + 1/2 x + c

= 6/4 sin (2/8u) + 1/2u + c

i don't know whether that's right, and i don't know how to do the rest.