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Calculus, EXT2 C1 2022 HSC 4 MC

Of the following expressions, which one need NOT contain a term involving a logarithm in its anti-derivative?

  1. `(x+2)/(x^(2)+4x+5)`
  2. `(x+2)/(x^(2)-4x-5)`
  3. `(x-1)/(x^(3)-x^(2)+x-1)`
  4. `(x+1)/(x^(3)-x^(2)+x-1)`
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`C`

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`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`

Calculus, EXT2 C1 2019 SPEC1-N 4

Evaluate  `int_(e^3) ^(e^4) (1)/(x log_e (x))\ dx`.   (3 marks)

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`log_e ((4)/(3))`

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`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))`

Calculus, EXT2 C1 2010 HSC 1b

Evaluate  `int_0^(pi/4) tan\ x\ dx`.   (3 marks) 

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`1/2 ln 2 \ \ text(or)\ \ ln\ sqrt2`

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`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`

Calculus, EXT2 C1 2013 HSC 14a

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)}`

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`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`