Of the following expressions, which one need NOT contain a term involving a logarithm in its anti-derivative?
- `(x+2)/(x^(2)+4x+5)`
- `(x+2)/(x^(2)-4x-5)`
- `(x-1)/(x^(3)-x^(2)+x-1)`
- `(x+1)/(x^(3)-x^(2)+x-1)`
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Of the following expressions, which one need NOT contain a term involving a logarithm in its anti-derivative?
`C`
`text{Consider the denominator of}\ C:`
`x^(3)-x^(2)+x-1` | `=x^2(x-1)+(x-1)` | |
`=(x^2+1)(x-1)` |
`(x-1)/(x^(3)-x^(2)+x-1)=(x-1)/((x^2+1)(x-1))=1/(x^2+1)`
`int 1/(x^2+1)\ dx=tan^(-1)(x)+c`
`=>C`
Evaluate `int_(e^3) ^(e^4) (1)/(x log_e (x))\ dx`. (3 marks)
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`log_e ((4)/(3))`
`text(Let)\ \ u = log_e x`
`(du)/(dx) = (1)/(x) \ => \ du = (1)/(x) dx`
`text(When) \ \ x = e^4 \ => \ u = 4`
`text(When) \ \ x = e^3 \ => \ u = 3`
`int_(e^3) ^(e^4) (1)/(x log_e (x))` | `= int_3 ^4 (1)/(u)\ du` |
`= [ log_e u]_3 ^4` | |
`= log_e 4 – log_e 3` | |
`= log_e ((4)/(3))` |
Find `int (ln x)/x\ dx.` (2 marks)
`((ln x)^2)/2 + c`
`text(Let)\ \ u=lnx,\ \ \ du=1/x\ dx`
`int (ln x)/x \ dx` | `=int u\ du` |
`=1/2 u^2 +c` | |
`=1/2 (ln x)^2 +c` |
Evaluate `int_0^(pi/4) tan\ x\ dx`. (3 marks)
`1/2 ln 2 \ \ text(or)\ \ ln\ sqrt2`
`int_0^(pi/4) tan\ x\ dx` | `=int_0^(pi/4) (sin\ x)/(cos\ x)\ dx` |
`=[-ln\ cos\ x]_0^(pi/4)` | |
`=[-ln\ cos\ pi/4 – (-ln cos 0)]` | |
`=-ln\ 1/sqrt2 + ln\ 1` | |
`=ln sqrt2` | |
`=1/2 ln 2` |
Evaluate `int_0^1 (e^(2x))/(e^(2x) + 1)\ dx`. (3 marks)
`1/2log_e((e^2 + 1)/2)`
`int_0^1 (e^(2x))/(e^(2x) + 1)\ dx` | `= 1/2[log_e(e^(2x) + 1)]_0^1` |
`= 1/2[log_e(e^2 + 1) − log_e2]` | |
`= 1/2log_e((e^2 + 1)/2)` |
The diagram shows the graph `y = ln x.`
By comparing relevant areas in the diagram, or otherwise, show that
`ln t > 2 ((t - 1)/(t + 1))`, for `t > 1.` (3 marks)
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`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Area under curve)` | `>\ text(Area of triangle)` |
`int_1^t ln x\ dx` | `> 1/2 xx (t – 1)ln t,\ \ \ t > 1` |
`underbrace{int_1^t 1\ln x\ dx}_text(integration by parts)` | `> 1/2 xx (t – 1)ln t` |
`[x ln x]_1^t – int_1^t x * 1/x\ dx` | `> ((t – 1)ln t)/2` |
`(tlnt – ln 1) – [x]_1^t` | `> ((t – 1)ln t)/2` |
`t ln t – (t – 1)` | `> ((t – 1) ln t)/2` |
`2t ln t – 2t + 2` | `> t ln t – ln t` |
`t ln t + ln t` | `> 2(t – 1)` |
`(t + 1) ln t` | `> 2 (t – 1)` |
`ln t` | `> 2 ((t – 1)/(t + 1)),\ \ \ t > 1` |