smc-1203-30-Log (Indefinite)

Calculus, 2ADV C4 2020 HSC 17

Find `int x/(4 + x^2)\ dx`. (2 marks)

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`1/2 log_e (4 + x^2) + c`

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`int x/(4 + x^2)\ dx`	`= 1/2 int (2x)/(4 + x^2)\ dx`
	`= 1/2 log_e (4 + x^2) + c`

Calculus, 2ADV C4 2004 HSC 3bii

Find `int x/(x^2-3)\ dx`. (2 marks)

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`1/2 ln (x^2 – 3) + C`

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	`int x/(x^2 – 3)\ dx`	`= 1/2 int (2x)/(x^2 – 3)\ dx`
		`= 1/2 ln (x^2 – 3) + C`

Calculus, 2ADV C4 2019 HSC 13c

Differentiate `(ln x)^2`. (2 marks)

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Hence, or otherwise, find `int(ln x)/x\ dx`. (1 mark)

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`(2 ln x)/x`
`1/2 (ln x)^2 + C`

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i.	`y`	`= (ln x)^2`
	`(dy)/(dx)`	`= 2 ⋅ 1/x ⋅ ln x`
		`= (2 ln x)/x`

♦ Mean mark part (ii) 49%.

ii.	`int (ln x)/x\ dx`	`=1/2 int (2 ln x)/x dx`
		`= 1/2 (ln x)^2 +C`

Calculus, 2ADV C4 2015 HSC 11h

Find `int x/(x^2 - 3)\ dx`. (2 marks)

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`1/2 ln (x^2 – 3) + C`

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`int x/(x^2 – 3)\ dx`

`= 1/2 int (2x)/(x^2 – 3)\ dx`

`= 1/2 ln (x^2 – 3) + C`

Calculus, 2ADV C4 2005 HSC 2ci

Find `int (6x^2)/(x^3 + 1)\ dx`. (2 marks)

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`2\ ln\ (x^3 + 1) +C`

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`int (6x^2)/(x^3 + 1)\ dx`	`= 2 int (3x^2)/(x^3 + 1)\ dx`
	`= 2 ln\ (x^3 + 1) + C`

Calculus, 2ADV C4 2008 HSC 2ci

Find `int (dx)/(x+5)`. (1 mark)

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`ln (x + 5) + C`

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`int (dx)/(x + 5)`

`= ln (x + 5) + C`

Calculus, 2ADV C4 2010 HSC 2dii

Find `intx/(4+x^2)\ dx`. (2 marks)

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`1/2ln(4+x^2)+C`

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`intx/(4+x^2)\ dx`

IMPORTANT: Minimise errors by adjusting the integral to fit the form `(f'(x))/(f(x))` before integrating.

`=1/2int(2x)/(4+x^2)\ dx`

`=1/2ln(4+x^2)+C`

Calculus, 2ADV C4 2012 HSC 12b

Find `int(4x)/(x^2+6)\ dx`. (2 marks)

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`2ln(x^2+6)+C`

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`int(4x)/(x^2+6)\ dx`

`=2int(2x)/(x^2+6)\ dx`

`=2ln(x^2+6)+C`