definite integral question?

Question

Evaluate the definite integral

first question.
8
∫(6x^2+2)/(x^(1/2))dx=?
1

second question.
5
∫(4x^2+1)/x^2dx=?
2

How do you start this problem?

Answer 1

You start by finding an antiderivative of each one (not difficult, just break up the fraction and use the power rule on each part). Then you evaluate it at the limits of integration, and subtract. For example:

[1, 8]∫(6x²+2)/√x dx
[1, 8]∫6x^(3/2) + 2x^(-1/2) dx
12/5 x^(5/2) + 4x^(1/2) from 1 to 8:
12/5 (8)^(5/2) + 4(8)^(1/2) - (12/5 (1)^(5/2) + 4(1)^(1/2))
1536√2/5 + 8√2 - (12/5 + 4)
1576√2/5 + 32/5

You can do the second one on your own.

Answer 2

i'm sorry yet Gianlino's evidence isn't valid. x^2 and sqrt(x) are inverse of one yet another, despite the indisputable fact that their integrals from 0 to at least a million are respectively a million/3 and a couple of/3 better frequently, the graphs of applications inverse of one yet another will be symmetrical wrt the first bisector. hence the integrals, measuring the area between those graphs and the x axis and the lines x=0 and x=a million, will frequently no longer be equivalent. they're usually so in straight forward words particularly cases. i imagine it really is amazingly basic. enable me have a attempt. Dina, your substitution is purely about the right one, look With u^3 = a million-x^7 J = Int(0,a million)(a million-x^7)^(a million/3) dx transforms as follows: (a million-x^7)^(a million/3) --> u dx --> d(a million-u^3)^(a million/7) So J = Int(a million,0)ud(a million-u^3)^(a million/7) = u(a million-u^3)^(a million/7)|_1^0 - Int_1^0 (a million-u^3)^(a million/7)du = 0 + your 2d critical so that they are equivalent ! you may want to apply an same trick for any (m,n) constructive integers in selection to (3,7) Edited: ok, my argument about gianlino's change into incorrect because i did not pay interest to his specifying reducing applications. And the replace of variables I proposed is precisely replacing function and variable (therefore making use of the inverse function) and is a demonstration of gianlino's declare. i choose some cafeine

Answer 3

1.the integral=[6(x^3/2)+2x^-1/2]dx
=6x^5/2 /5/2+2x^1/2 /1/2
=(12/5)x^5/2+4x^1/2
now apply the limits
(12/5)[8^5/2-1^5/2]+4[8^1/2-1^1/2]

2.integral=[4+x^-2]dx
=4x-1/x
apply the limits
=4[5-2]-[1/5-1/2]