integration question involving using a substitution?

Question

hi. i would like to find out how to solve this problem.

i would like to use a substitution to determine

∫ dx / (2x+2)^2

the dx is divided by 2x+2 squared

Answer 1

say 2x +2 = t
the derivative is
2dx = dt
dx = dt/2
∫ dx / (2x+2)^2 changes into
∫ dt/2t^2 = 1/2*(-1/t)
than you go back to x and you get
-1/2*1/(2x+2)

Answer 2

substitute using u=2x+2, then you'll have du=2dx, substituting in the original you'll have to integrate (u^-2)/2 with du, making it simple (you'll get (-1/2)(u^-1) and substituting u, you'll get (-1/2)(2x+2)^-1, the integrated function

Answer 3

? (0 to a million, a million/(a million + x^2)^2 dx ) enable x = tan(t). Then dx = sec^2(t) dt Calculate our new bounds. whilst x = 0, then 0 = tan(t), so t = 0. whilst x = a million, then a million = tan(t), so t = pi/4 That makes our new bounds 0 to pi/4. ?(0 to pi/4, a million/(a million + tan^2(t))^2 sec^2(t) dt ) save on with the identity a million + tan^2(t) = sec^2(t). ?(0 to pi/4, a million/(sec^2(t))^2 sec^2(t) dt ) sq. it. ?(0 to pi/4, a million/sec^4(t) sec^2(t) dt ) And we get a alleviation. ?(0 to pi/4, sec^2(t)/sec^4(t) dt ) ?(0 to pi/4, a million/sec^2(t) dt ) ?(0 to pi/4, cos^2(t) dt ) Use the identity cos^2(t) = (a million/2)(a million + cos(2t)) ?(0 to pi/4, (a million/2)(a million + cos(2t)) dt ) element the consistent (a million/2). (a million/2) ?(0 to pi/4, (a million + cos(2t)) dt ) And now the required is easy. (a million/2) ( t + (a million/2)sin(2t)) {evaluated from 0 to pi/4} (a million/2) ( [ pi/4 + (a million/2)sin(2*pi/4) ] - [ 0 + (a million/2)sin(2*0) ] ) Simplify. (a million/2) ( [ pi/4 + (a million/2)(pi/2) ] - [ 0 + (a million/2)(0) ] ) (a million/2) ( [pi/4 + pi/4] - 0 ) (a million/2) (2pi/4) (a million/2) (pi/2) pi/4

Answer 4

I = (1/4) ∫ 1 / (x + 2)² dx
Let u = x + 2
du = dx
I = (1/4) ∫ (1 / u²).du
I = (1/4) ∫ u^(-2).du
I = - (1/4).u^(-1) + C
I = - 1 / (4.(x + 2)) + C