Hello!
I've got some really tough question and I could use some help with it.
calculate the result and choose the closest answer to these given answers
The answers are:
A -450.3091514312
B 450.3091514312
C -45913.2569590715
D 45913.2569590715
E -6.5458518477
F 6.5458518478
G -0.1336527864
H 0.1336527864
I The result deosn't belong to real number group, it belongs to complex numbers group.
K The result inclines to infinity.
Could anybody give me some help?
I see 3 integrations, but is this in fact one problem? It appears that the integrations above and below actually represent the limits of integration to the one in the middle. Am I correct in assuming this?
The bottom integration does not have an obvious dx. Is the "d[x^2+3]" suposed to be the derivative of x^2+3, where d/dx[x^2+3]=2x*dx?
If so, then the bottom integration becomes:
int(0:0.125)[4^x^2*2x]dx ... let u=x^2, then du=2xdx
int(0:0.125)[4^u]du = (1/ln4)*4^u ... back substitute u=x^2
(1/ln4)[4^x^2] from (0:0.125) = (1/ln4){[1]-[4^0.125^2]} = (1/ln4){-.02189714865) = (.72134752044)*(-.02189712865) = -.01579545388
AfterShock is right about the top integration being 0, so now we have the lower and upper limits of integration on the middle integration.
int(-.01579545388:0)[(2a-10)^3/sqrt(1+5a)]da
Integrating that might not prove as easy as the other two integrations.
Here goes:
let 1+5a=u, which means that a = 1/5(u-1) ... (if 1+5a=u is solved for a) then 5da=du, which means that da=(1/5)du
1/5*int(-.01579545388:0){[2(1/5(u-1))-10]^3/sqrt(u)}du
1/5*int(-.01579545388:0){[0.4u-0.4-10]^3/sqrt(u)}du
1/5*int(-.01579545388:0){[0.4u-10.4]^3/sqrt(u)}du
Let's ignore the integration for a second and simplify the fraction:
(0.4u-10.4)^3/sqrt(u) ... multiply the numerator
[ (0.4u)^3 + 3*(0.4u)^2*(-10.4) + 3(0.4u)*(-10.4)^2 + (-10.4)^3 ] / sqrt(u)
[ 0.064u^3 - 31.2*.16u^2 + .4u*108.16 - 1124.864 ] / sqrt(u)
[ 0.064u^3 - 4.992u^2 + 43.264u - 1124.864 ] / sqrt(u)
Now, going back to the integration:
int(-.01579545388:0){[0.064u^3-4.992u^2+43.264u-1124.864] / sqrt(u)}da ... separate the numerators
int(-.01579545388:0)[0.064u^3/sqrt(u)-4.992u^2/sqrt(u)+43.264u/sqrt(u)-1124.864/sqrt(u)]da ... the division of u's can be simplified by subtracting their exponents
int(-.01579545388:0)[0.064u^(2.5)-4.992u^(1.5)+43.264u^(0.5)-1124.864^(-0.5)]da ... now integrate, were integrating u^n = 1/(n+1)*u^(n+1)
[(1/3.5)0.064u^(3.5)-(1/2.5)4.992u^(2.5)+(1/1.5)43.264u^(1.5)-(1/0.5)1124.864^(0.5)] from (-.01579545388:0)
Now, simplify and plug in the limits of integration. I would, but there are too many number to keep track of . Good luck!
Oh, I see, I didn't recogize the upper and lower integrals to be the limits of integration of the middle integral.
Using Maple to calculate the integrals and simplify, it got:
5.334874
The calculation was done as follows:
int((2*a-10)/sqrt(1+5*a), a = int(2*4^(x^2+3)*x, x = 0 .. 0.125) .. 0)