smc-965-30-Indefinite integrals

Calculus, 2ADV C2 2023 MET1 1a

Let \(y=\dfrac{x^2-x}{e^x}\).

Find and simplify \(\dfrac{dy}{dx}\). (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

\(\dfrac{-x^2+3x-1}{e^x}\)

Show Worked Solution

\(\text{Using the quotient rule:}\)

\(\dfrac{dy}{dx}\)	\(=\dfrac{e^x(2x-1)-(x^2-x)e^x}{(e^x)^2}\)
	\(=\dfrac{e^x(-x^2+3x-1)}{e^{2x}}\)
	\(=\dfrac{-x^2+3x-1}{e^x}\)

Calculus, 2ADV C4 2020 HSC 4 MC

What is `int e + e^(3x)\ dx`?

`ex + 3e^(3x) + c`
`ex + 1/3e^(3x) + c`
`e + 3e^(3x) + c`
`e + 1/3e^(3x) + c`

Show Answers Only

`B`

Show Worked Solution

`int e + e^(3x) = ex + 1/3e^(3x) + c`

`(text(Note)\ e\ text(is a simple constant here))`

`=> B`

Calculus, 2ADV C4 2006 HSC 2bi

Find `int 1 + e^(7x)\ dx`. (2 marks)

Show Answers Only

`x + 1 /7 e^(7x) + c`

Show Worked Solution

`int 1 + e^(7x) \ dx = x + 1/7 e^(7x) + c`

Calculus, 2ADV C4 2014 HSC 4 MC

Which expression is equal to `int e^(2x)\ dx`?

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

Show Answers Only

`C`

Show Worked Solution

`int e^(2x)\ dx`

`= 1/2 e^(2x) + C`

`=> C`

Calculus, 2ADV C4 2013 HSC 11e

Find `inte^(4x+1)dx` (2 marks)

Show Answer Only

`1/4e^(4x+1)+C`

Show Worked Solutions

`inte^(4x+1)\ dx=1/4e^(4x+1)+C`