smc-1037-30-Sin/Cos Integration

Calculus, EXT1 C2 2023 HSC 11d

Find \( {\displaystyle \int} \dfrac{1}{\sqrt{4-9x^2}}\ dx\) (2 marks)

Show Answers Only

\(\dfrac{1}{3} \sin^{-1} \Big{(}\dfrac{3x}{2} \Big{)} +c \)

Show Worked Solution

\({\displaystyle \int} \dfrac{1}{\sqrt{4-9x^2}}\ dx\)	\(=\dfrac{1}{3} {\displaystyle \int} \dfrac{3}{\sqrt{2^2-(3x)^2}}\ dx\)
	\(=\dfrac{1}{3} \sin^{-1} \Big{(}\dfrac{3x}{2} \Big{)} +c \)

Calculus, EXT1 C2 2021 HSC 11f

Evaluate `int_0^sqrt3 1/(sqrt(4 - x^2))\ dx`. (2 marks)

Show Answers Only

`pi/3`

Show Worked Solution

	`int_0^sqrt3 1/(sqrt(4 – x^2))\ dx`	`= [sin^(-1)\ x/2]_0^sqrt3`
		`= sin^(-1)\ sqrt3/2 – sin^(-1) 0`
		`= pi/3`

Calculus, EXT1 C2 2004 HSC 1d

Find `int_0^1(dx)/(sqrt(4 − x^2))`. (2 marks)

Show Answers Only

`pi/6`

Show Worked Solution

`int_0^1(dx)/(sqrt(4 − x^2))`	`= [sin^(−1)\ x/2]_0^1`
	`= sin^(−1)(1/2) − sin^(−1)(0)`
	`= pi/6`

Calculus, EXT1 C2 2015 HSC 7 MC

What is the value of `k` such that `int_0^k 1/sqrt(4 − x^2) \ dx= pi/3 ?`

`1`
`sqrt3`
`2`
`2sqrt3`

Show Answers Only

`B`

Show Worked Solution

`int_0^1 1/sqrt(4 − x^2)dx`	`= pi/3`
`[sin^(−1)\ x/2]_0^k`	`= pi/3`
`sin^(−1)\ k/2 − sin^(−1)\ 0`	`= pi/3`
`sin^(−1)\ k/2`	`= pi/3`
`k/2`	`= sin\ pi/3`
	`= sqrt3/2`
`:.k`	`= sqrt3`

`⇒ B`

Calculus, EXT1 C2 2008 HSC 1c

Evaluate `int_-1^1 1/sqrt(4 - x^2)\ dx`. (2 marks)

Show Answers Only

`pi/3`

Show Worked Solution

`int_-1^1 1/sqrt(4 – x^2)\ dx`

`= [sin^(-1) (x/2)]_(-1)^1`

`= sin^(-1) (1/2) – sin^(-1) (-1/2)`

`= pi/6 – (- pi/6)`

`= pi/3`

Calculus, EXT1 C2 2013 HSC 11b

Find `int 1/sqrt (49 - 4x^2)\ dx`. (2 marks)

Show Answers Only

`1/2 sin^(-1) ((2x)/7) + c`

Show Worked Solution

`int 1/sqrt(49 – 4x^2)\ dx`

`= int 1/(2 sqrt(49/4 – x^2))\ dx`

`= 1/2 int 1/sqrt((7/2)^2 – x^2)\ dx`

`= 1/2 sin^(-1) ((2x)/7) + c`

Calculus, EXT1 C2 2010 HSC 1a

Use the table of standard integrals to find `int 1/sqrt(4 - x^2)\ dx`. (1 mark)

Show Answers Only

`sin^(-1)\ x/2 + c`

Show Worked Solution

`int 1/sqrt(4\ – x^2)\ dx`

`= sin^(-1)\ x/2 + c`