Integration and the logarithmic function (1 Viewer)

[luvotomy]

i'd really appreciate it if someone could help me with these very annoying questions:

1) Find the exact area between the curve y=1/x, the x-axis and the lines y=x and x=2 in teh first quadrant.

ans: (0.5+ln2) units^2

2) Find the indefinite integral (primitive function) of 3^2x-1.

ans: 3^2x-1/2ln3 +C

 

[lolokay]

for 2,

3^2x-1 = e^u
where u = (2x-1)ln3
now du/dx = 2 ln3
so dx/du = 1/2ln3
Int 3^2x-1 .dx = 1/2ln3 Int e^u .du
= 1/2ln3 e^u + C
= 3^2x-1/2ln3 + C

 

[Trebla]

1) Finding the intersection of y = x and y = 1/x
1/x = x
x² = 1
x = -1,1
but since we're concerned with the first quadrant, we take x = 1
Now if you draw a diagram, the area is split into two parts. One part is below the line y = x only and the other is below y = 1/x only
So we have A = ∫¹₀x dx + ∫²₁(1/x) dx
= [x²/2]¹₀ + [ln x]²₁
= 1/2 + ln 2