Q27 of 29 Page 1

Evaluate the following:

OR


Evaluate as the limit of a sum.

Let


Using property




Add (i) and (ii)






We know that cos 2x = 1 – 2sin2x




Divide by cos2x in numerator and denominator




We know that 1 + tan2x = sec2x




Let tan x = t


The limits will also change


When



and





So, the limits will be -1 to 1


Now,



⇒ sec2x dx = dt


Hence





We know that







Hence


OR


Here f(x) = 3x2 – 2x + 4


Expressing integral as limit of sum



Where





Let us find f (a + rh)


⇒ f(a + rh) = 3(a + rh)2 – 2(a + rh) + 4


⇒ f(a + rh) = 3(a2 + 2arh + r2h2) – 2a – 2rh + 4


⇒ f(a + rh) = 3a2 + 6arh + 3r2h2 – 2a – 2rh + 4


Here a = -2 and b = 2


b – a = 2 – (-2) = 4 and





Put this value of f(a + rh) in (i)




Since ,




And,











Put the limit



As



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