some integration questions?

Question

using substitution..

1) ∫cos x/(2sinx+5)^3 dx
2) ∫1/t^2 e^(2/t) dx
3) ∫(2+tan x)/cos^2 x dx

and btw how do i know when to use integration by sub, parts or standard?

Answer 1

1) ∫cos x / (2 sin x + 5)^3 dx: let u = 2 sin x + 5, du = 2 cos x dx
= ∫(1/2) du / u^3
= (1/2) u^(-2) / (-2) + c
= -1 / (4(2 sin x + 5)^2) + c.

2) ∫(1/t^2) e^(2/t) dt: let u = 1/t, du = -1/t^2 dt
= ∫-e^(2u) du
= -e^(2u) / 2 + c
= -e^(2/t) / 2 + c.

3) ∫(2 + tan x) / cos^2 x dx
= ∫(2 + tan x) sec^2 x dx: let u = 2 + tan x, du = sec^2 x dx
= ∫u du
= u^2 / 2 + c
= (2 + tan x)^2 / 2 + c.

Use substitution if you can replace an awkward term with a simpler one and the derivative of the substituting expression appears in the integrand (or something very much like it).

Use integration by parts mainly when there's no good substitution and the integral will become simpler if you differentiate one part.

Answer 2

basically there in no set rule where u have to use sub or parts or standard
where ever u have a prob just try to reduce it into a form which can be integrated easily. for that u should try to practice a lot .

1)
take 2sinx+5=t
diffrentiating both sides u get
2cosx(dx)=dt;
cosx(dx)=dt/2;
putting this in ur intgral.
∫cos x/(2sinx+5)^3 dx=

∫(1/t^3)dt/2= -(1/t^2)*(1/4)
in ur ans put t=2sinx+5.

Answer 3

⌠cos x /(2 sin x + 5)^3 dx = 1/2⌠(2 sin x + 5)^-3 (2cos x) dx
= 1/2 (2 sin x + 5)^-2 / (-2 ) + c
= - 1 / [ 4 (2 sin x + 5 )^2 ] + c

⌠ 1/t^2 e^(2/t) dx
= -1/2 ⌠ e^(2/t) (-2/t^2) dx
= - 1/2 e^(2/t) + c

⌠ (2+ tan x) / ( cos^2 x ) dx
=