Evaluate the integrals

Hi there, I am working on a review test to study for a test, and I need help on the following problems:

Evaluate the integral dx/((x^2 + 2x + 2)^2)

and

Evaluate the integral sqrt (e^(2t) - 16)dt

please explain the steps. Thanks!

dx / (x^2 + 2x + 2)^2 =>
dx / (x^2 + 2x + 1 + 1)^2 =>
dx / ((x + 1)^2 + 1)^2

u = x + 1
du = dx

du / (1 + u^2)^2

u = tan(t)
du = sec(t)^2 * dt

sec(t)^2 * dt / (1 + tan(t)^2)^2 =>
sec(t)^2 * dt / (sec(t)^2)^2 =>
sec(t)^2 * dt / sec(t)^4 =>
dt / sec(t)^2 =>
cos(t)^2 * dt =>
cos(2t/2)^2 * dt =>
(1/2) * (1 + cos(2t)) * dt

Integrate

(1/2) * (t + (1/2) * sin(2t)) + C =>
(1/4) * (2t + sin(2t)) + C

u = tan(t)
t = arctan(u)

u = sin(t)/cos(t)
u * cos(t) = sin(t)

(1/4) * (2 * arctan(u) + 2 * sin(t) * cos(t)) + C =>
(1/4) * 2 * (arctan(u) + u * cos(t) * cos(t)) + C =>
(1/2) * (arctan(u) + u * cos(t)^2) + C =>
(1/2) * (arctan(u) + u / sec(t)^2) + C =>
(1/2) * (arctan(u) + u / (1 + tan(t)^2)) + C =>
(1/2) * (arctan(u) + u / (1 + u^2)) + C =>

u = x + 1

(1/2) * (arctan(x + 1) + (x + 1) / (1 + (x + 1)^2)) + C =>
(1/2) * arctan(x + 1) + (x + 1) / (2 * (x^2 + 2x + 2)) + C =>
arctan(x + 1) / 2 + (x + 1) / (2x^2 + 4x + 4) + C




sqrt(e^(2t) - 16) * dt
sqrt(e^(2t) - 16) * dt * e^t / e^t

e^t = 4 * sec(x)
e^t * dt = 4 * sec(x) * tan(x) * dx

sqrt(16 * sec(x)^2 - 16) * 4 * sec(x) * tan(x) * dx / (4 * sec(x))
4 * tan(x) * tan(x) * dx =>
4 * tan(x)^2 * dx =>
4 * (sec(x)^2 - 1) * dx

Integrate

4 * (tan(x) - x) + C

e^t = 4 * sec(x)
(1/4) * e^t = sec(x)
(1/16) * e^(2t) = sec(x)^2
(1/16) * e^(2t) = tan(x)^2 + 1
(e^(2t) - 16) / 16 = tan(x)^2
(1/4) * sqrt(e^(2t) - 16) = tan(x)

x = arcsec((1/4) * e^(t))

4 * (tan(x) - x) + C =>
4 * (1/4) * sqrt(e^(2t) - 16) - 4 * arcsec((1/4) * e^(t)) + C =>
sqrt(e^(2t) - 16) - 4 * arcsec((1/4) * e^(t)) + C