Please Solve these integrals, I have the answers just dont understand the steps.?

Question

1)
integrand ((Cos(x))/(1+((sin(x))^2))dx)
upper limit pi/2 and lower limit 0

2)
integrand ((e^(-x))/(1+(e^(-2x)))dx)

Please show your steps.
Answers
1) pi/4
2) -arctan(e^(-x))+c (c is a constant)

BTW... Thanx in advance

Answer 1

1) say sin x=t
-cos x dx=dt
integrand (-dt/(1+t^2)) with upper lim 1 and lower lim 0

2)it's the same
e^(-x)=t
-e^(-x)dx=dt
integrand (-dt/(a+t^2))

Answer 2

Writing a minimum of a few words of the sequence for a million/ln x should not be a topic. although the place you elect to amplify your imperative is important. additionally the imperative you're talking approximately is improper if x is larger than a million, as lim (x->a million) (a million/lnx)=+-infty. Now my wager is that if the imperative is well-known to be non-elemantary then getting a formulation for its taylor advance would desire to be very confusing if no longer impossible. that's fairly helpful to take a seem at numeric integration books for the form to handle questions like this.

Answer 3

1.
I = ∫ cos x / (1 + sin x)² dx----- lims 0 and π/2

Let u = sinx

du = cos x dx

when x = 0, u = 0
when x = π/2, u = 1

I = ∫ 1/(1 + u²) du---------- lims 0 and 1

I = tan^(-1) u ---------- lims 0 and 1

I = π / 4


2
I = ∫ e^(-x) / ((1 + e^(-2x)) dx

Let y = e^(-x)

y² = e^(-2x)

dy / dx = - e^(-x)

- dy / dx = e^(-x)

- dy = e^(-x) dx

I = - ∫ dy / (1 + y²)

I = - tan^(-1) y + C

I = - tan ^(-1) e^(-x) + C

Answer 4

1) Integrate ∫{cos(x)/[1 + sin²(x)]}dx
Let
u = sin(x)
du = cos(x)dx

∫{cos(x)/[1 + sin²(x)]}dx
= ∫{1/(1 + u²)}du
Let
tan(θ) = u
sec²(θ)dθ = du
= ∫{1/(1 + u²)}du = ∫{sec²(θ)/(1 + tan²(θ)}dθ
= ∫{sec²(θ)/(sec²(θ)}dθ
= ∫dθ = θ = arctan(u) = arctan(sin(x)) {evaluated from 0 to π/2}
= arctan(sin(π/2)) - arctan(sin(0)) = π/4 - 0 = π/4
________________

2) Integrate ∫{e^(-x)/[1 + e^(-2x)]}dx
u = e^(-x)
du = -e^(-x)dx

∫{e^(-x)/[1 + e^(-2x)]}dx = -∫{1/(1 + u²)}du
Let
tan(θ) = u
sec²(θ)dθ = du
= -∫{1/(1 + u²)}du = -∫{sec²(θ)/(1 + tan²(θ)}dθ
= -∫{sec²(θ)/(sec²(θ)}dθ
= -∫dθ = -θ + C = -arctan(u) + C = -arctan(e^(-x)) + C