Find the integral? ∫ dx / ( 9 + 2x²)x3/2?

Question

∫ dx / ( 9 + 2x²)3/2

Answer 1

(2/3) ∫ 1/(9+2x²) dx = (1/3) ∫ 1/(9/2 + x²) dx

Recall that d/dx atan(x) = 1/(1+x²)

Let au = x ; a du = dx

∫ 1/(a² + x²) dx = ∫ adu/(a² + a²u²) = (1/a ) ∫ du / (1 + u²)  = 1/a atan(u) + c

∫ 1/(a² + x²) dx = 1/a atan(x/a) + c

(1/3) ∫ 1/(9/2 + x²) dx = (1/3) √2 / 3 atan (x √2/3 ) + C = √2/9 atan(x √2/3 ) + C

Answer: √2/9 atan(x √2/3 ) + C

Answer 2

quite interesting....

∫ dx / (9 + 2x²)^(3/2) =

∫ dx / √(9 + 2x²)³ =

∫ dx / [(9 + 2x²)√(9 + 2x²)] =

let x = (3/√2) tan u →

dx =  (3/√2) sec²u du

thus, substituting, you get:

∫ dx / [(9 + 2x²)√(9 + 2x²)] = ∫ (3/√2)sec²u du/ {9 + 2[(3/√2)tan u]²}√{9 + 2[(3/√2)tan u]²} =

∫ (3/√2) sec²u du / {9 + 2[(9/2) tan²u]}√{9 + 2[(9/2) tan²u]} =

∫ (3/√2) sec²u du / [(9 + 9 tan²u)√(9 + 9 tan²u)] =

factoring out 9:

∫ (3/√2) sec²u du / {9(1 + tan²u)√[9(1 + tan²u)]} =

∫ (3/√2) sec²u du / [9(1 + tan²u) 3√(1 + tan²u)] =

according to trig identity, replace 1 + tan²u with sec²u:

∫ (3/√2) sec²u du / [27sec²u √(sec²u)] =

∫ (3/√2) sec²u du / [27sec²u (sec u)] =

3 sec²u canceling out,

∫ (1/√2) du / (9 sec u) =

taking the constants out,

[1/(9√2)] ∫  (1 / sec u) du =

[1/(9√2)] ∫ cos u du =

[1/(9√2)] sin u + C

to sum up:

x = (3/√2) tan u → tan u = (x√2)/3

then, finding out sin u in terms of tan u, you get:

1/sin²u = csc²u = 1 + cot²u = 1 + 1/tan²u = [(tan²u + 1)/tan²u]

thus

sin²u =  [tan²u/(tan²u + 1)] → |sin u| = |tan u| /√(tan²u + 1) →

sin u = tan u /√(tan²u + 1) →

therefore, being tan u = (x√2)/3,

sin u = [(x√2)/3] /√ {[(x√2)/3]² + 1} = [(x√2)/3] /√ [(2x²/9) + 1] =

[(x√2)/3] /√[(2x²+ 9)/ 9] = [(x√2)/3] [3/√(2x²+ 9)] = [(x√2)/√(2x²+ 9)]

and finally substituting back, you get:

∫ dx / (9 + 2x²)^(3/2) = [1/(9√2)] [(x√2)/√(2x²+ 9)] + C =  (1/9)x (2x²+ 9)^(-1/2) + C

I hope it helps..

Bye!

Answer 3

i take it you are intergrating with respect to x;

=int.3(9+2x^2)/2  dx

=int.27+6x^2  dx

=27x+(6/3)x^3 + c

=27x+2x^3+c

hope this helps

Answer 4

that's tough. math is one of my weakest subjects.

Answer 5

do your own homework duh

answer is = you might be a floogle stoofle  

Answer 6

i have no clue