4Units Integration (1 Viewer)

[conman]

Can some one help me to solve the following questions:

Also including final answers

 

[kony]

first one, subsitute u² = 2-x.

still thinking about the others.

 

[kony]

the third one seems to be some sort of combination of by parts and substitution, but i'm not sure.

edit: okay, q2 and q3 are from the special properties of integrals, namely the one where the integral of f(x) from 0 to a is equal to the integral of f(a-x) from 0 to a.

(use subsitution at the end of my working)

 

[conman]

Sorry to mention, u can use futher properties of definite integral to solve all these questions. I just can get solutions right as in text book. The last question I dont know how to prove inequality

 

[elseany]

what text book are these coming from?

 

[AMorris]

Lovely questions . If you try putting either 2 or 3 into the Integrator (integrate.wolfram.com) you get a huge expression which is impossible to evaluate, but the using the substitution makes them so much nicer.

 

[Trebla]

The inequality can be obtained using the fact that 0 ≤ sin²x ≤ 1
.: 0 ≤ 16sin²x ≤ 16
9 ≤ 9 + 16sin²x ≤ 25
As both ends of the inequality are positive we can take the square root of everything to obtain the result.
3 ≤ √(9 + 16sin²x) ≤ 5
The idea of upper and lower rectangles is not necessary to obtain the final result. As the function restricted in the interval is positive, simply integrate everything from 0 to π/2 with respect to x to get the result.
i.e. ∫3 dx ≤ ∫√(9 + 16sin²x) dx ≤ ∫5 dx, all with limits from 0 to π/2, hence
3π/2 ≤ ∫√(9 + 16sin²x) dx ≤ 5π/2