smc-1193-15-tan

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?

`(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)`

Show Answers Only

`C`

Show Worked Solution

`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 2021 SPEC2 2

Evaluate `int_0^1 (2x + 1)/(x^2 + 1)\ dx`. (3 marks)

Show Answers Only

`text(See Worked Solutions)`

Show Worked Solution

`int_0^1 (2x + 1)/(x^2 + 1)\ dx`	`= int_0^1 (2x)/(x^2 + 1)\ dx + int_0^1 1/(x^2 + 1)\ dx`
	`= [log_e(x^2 + 1)]_0^1 + [tan^(-1)(x)]_0^1`
	`= log_e 2 – log_e 1 + tan^(-1)(1) – tan^(-1)(0)`
	`= log_e 2 + pi/4`

Calculus, EXT2 C1 2021 HSC 12a

Find `int {2x + 3}/{x^2 + 2x + 2} dx`. (3 marks)

Show Answers Only

`ln (x^2 + 2x + 2) + tan^{-1} (x + 1) + c`

Show Worked Solution

`int {2x + 3}/{x^2 + 2x + 2} dx`	`= int {2x + 2}/{x^2 + 2x + 2} dx + int {1}/{(x – 1)^2 + 1} dx`
	`= ln (x^2 + 2x + 2) + tan^{-1} (x + 1) + c`

Calculus, EXT2 C1 2020 HSC 6 MC

Which expression is equal to `int frac{1}{x^2 + 4x + 10}\ dx`?

`frac{1}{sqrt(6)} tan^-1 (frac{x + 2}{sqrt{6} )) + c`
`tan^-1 (frac{x + 2}{sqrt{6} )) + c`
`frac{1}{2 sqrt(6)} ln | frac{x + 2 - sqrt(6)}{x + 2 + sqrt(6)} | + c`
`ln | frac{x + 2 - sqrt(6)}{x + 2 + sqrt(6)} | + c`

Show Answers Only

`A`

Show Worked Solution

`int frac{1}{x^3 + 4x + 10}\ dx`	`= int frac{1}{(x + 2)^2 + (sqrt6)^2}\ dx`
	`= frac{1}{6} tan^-1 (frac{x + 2}{sqrt6}) + c`

`=> \ A`

Calculus, EXT2 C1 SM-Bank 1

By completing the square and using the table of standard integrals, find

`int(dx)/(4x^2-4x+10)` (2 marks)

Show Answers Only

`1/6 tan^-1((2x-1)/(3))+C`

Show Worked Solution

`int(dx)/(4x^2-4x+10)`	`=int(dx)/(3^2+(2x-1)^2)`
	`=1/2 int 2/(3^2+(2x-1)^2) \ dx`
	`=1/6 tan^-1((2x-1)/(3))+C`

Calculus, EXT2 C1 2019 HSC 11c

Find `int (dx)/(x^2 + 10x + 29)` (2 marks)

Show Answers Only

`1/2 tan^(-1) ((x + 5)/2) + C`

Show Worked Solution

`int (dx)/(x^2 + 10x + 29)`	`= int (dx)/((x + 5)^2 + 2^2)`
	`= 1/2 tan^(-1) ((x + 5)/2) + C`

Calculus, EXT2 C1 2019 HSC 7 MC

Which of these integrals has the largest value?

`int_0^(pi/4) tan x\ dx`
`int_0^(pi/4) tan^2 x\ dx`
`int_0^(pi/4) 1 - tan x\ dx`
`int_0^(pi/4) 1 - tan^2 x\ dx`

Show Answers Only

`D`

Show Worked Solution

`text(Consider options A and B:)`

`text(Consider options C and D:)`

`:. int_0^(pi/4) 1 – tan^2 x\ dx\ \ text(is the largest)`

`text{(largest area under the curve}`

`text(between)\ \ x=0 \ and \ x=pi/4. text{)}`

`=> D`

Calculus, EXT2 C1 2018 HSC 12c

Find `int(x^2 + 2x)/(x^2 + 2x + 5)\ dx`. (3 marks)

Show Answers Only

`x – 5/2 tan^(−1) ((x + 1)/2) + c`

Show Worked Solution

`int(x^2 + 2x)/(x^2 + 2x + 5)\ dx`	`= int((x^2 + 2x + 5) – 5)/(x^2 + 2x + 5)\ dx`
	`= int 1 – 5/(x^2 + 2x + 5)\ dx`
	`= int 1 – 5/(2^2 + (x + 1)^2)\ dx`
	`= x – 5/2 tan^(−1) ((x + 1)/2) + c`

Calculus, EXT2 C1 2007 HSC 1b

Find `int tan^2 x sec^2 x\ dx.` (2 marks)

Show Answers Only

`1/3 tan^3 x + c`

Show Worked Solution

`int tan^2 x sec^2 x\ dx = 1/3 tan^3 x + c`

Calculus, EXT2 C1 2013 HSC 1 MC

Which expression is equal to `int tan x\ dx?`

`sec^2 x + c`
`-ln (cos x) + c`
`(tan^2 x)/2 + c`
`ln (sec x + tan x) + c`

Show Answers Only

`B`

Show Worked Solution

`int tan x\ dx =`	`int (sin x)/(cos x)\ dx`
`=`	`-ln (cos x) + c`

`=> B`

Calculus, EXT2 C1 2006 HSC 1b

By completing the square, find

`int (dx)/(x^2 - 6x + 13)` (2 marks)

Show Answers Only

`1/2 tan^-1 ((x – 3)/2) + c`

Show Worked Solution

`int (dx)/(x^2 – 6x + 13)`	`=int (dx)/((x – 3)^2 + 4)`
	`=1/2 tan^-1 ((x – 3)/2) + c`

Calculus, EXT2 C1 2009 HSC 1c

Find `int x^2/(1 + 4x^2)\ dx.` (3 marks)

Show Answers Only

`x/4 – 1/8 tan^-1 2x + c`

Show Worked Solution

`x^2/(1 + 4x^2)`	`= 1/4 xx (4x^2)/(1 + 4x^2)`
	`= 1/4 xx (1 + 4x^2)/(1 + 4x^2) – 1/4 xx 1/(1 + 4x^2)`
	`=1/4-1/4 xx 1/(1 + 4x^2)`

`:.int x^2/(1 + 4x^2)\ dx`	`= 1/4 int 1\ dx – 1/4 int 1/(1 + 4x^2)\ dx`
	`= x/4 – 1/4 xx 1/2 tan^-1 2x + c`
	`= x/4 – 1/8 tan^-1 2x + c`

Calculus, EXT2 C1 2010 HSC 1b

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

Show Answers Only

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

Show Worked Solution

`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 2011 HSC 1e

Evaluate `int_-1^1 1/(5 - 2t + t^2) \ dt.` (3 marks)

Show Answers Only

`pi/8`

Show Worked Solution

`int_-1^1 1/{(5 – 2t + t^2)}dt`	`= int_-1^1 1/{(4 + 1 – 2t + t^2)dt}`
	`= int_-1^1 1/(4 + (t – 1)^2)dt`
	`= 1/2[tan^-1 ((t – 1)/2)]_-1^1`
	`= 1/2 [tan^-1 0 – tan^(-1)(-1)]`
	`= 1/2 [0 – (-pi/4)]`
	`= pi/8`

Calculus, EXT2 C1 2012 HSC 11c

By completing the square, find `int (dx)/(x^2 + 4x + 5)`. (2 marks)

Show Answers Only

`tan^(−1)\ (x + 2) + c`

Show Worked Solution

`int (dx)/(x^2 + 4x + 5)`	`= int (dx)/(x^2 + 4x + 4 + 1)`
	`= int (dx)/((x + 2)^2 + 1)`
	` = tan^(−1)\ (x + 2) + c`