Please can you help with the following:
Q1)
find dy/dx when y=(5x² + 2)³(3x - 1)²
Q2)
with the help of integration by parts find:
∫ (2x+1) ln x dx
many thanks for any help given!
Luke.
Luke, If I were you, I would go to this website Wolfram|Alpha
In the input window, type in this exact phrase: integrate integrate (2x+1)ln[x]dx. (you can copy & paste)
Click the equals bar at the end if the input window.
Be sure you click ‘show steps’ to see the solution.
You can also use it to find: differentiate (5x^2 +2)^3 (3x-1)^2
Q1. You can either expand everything out, or use a combination of the chain and product rules.
$\displaystyle y = (5x^2 + 2)^3(3x - 1)^2$
$\displaystyle \frac{dy}{dx} = (5x^2 + 2)^3\,\frac{d}{dx}[(3x - 1)^2] + (3x - 1)^2\,\frac{d}{dx}[(5x^2 + 2)^3]$
$\displaystyle = 6(3x - 1)(5x^2 + 2)^3 + 15(3x - 1)^2(5x^2 + 2)^2$
$\displaystyle = 3(3x - 1)(5x^2 + 2)^2[2(5x^2 + 2) + 5(3x - 1)]$
Q2. $\displaystyle \int{(2x + 1)\ln{x}\,dx}$.
Remember that $\displaystyle \int{u\,dv} = uv - \int{v\,du}$
Let $\displaystyle u = \ln{x}$ so $\displaystyle du = \frac{1}{x}$
Let $\displaystyle dv = 2x + 1$ so $\displaystyle v = x^2 + x = x(x + 1)$.
So $\displaystyle \int{(2x + 1)\ln{x}\,dx} = x(x + 1)\ln{x} - \int{x(x + 1)\frac{1}{x}\,dx}$
$\displaystyle = x(x + 1)\ln{x} - \int{x + 1\,dx}$
$\displaystyle = x(x + 1)\ln{x} - x^2 - x + C$
$\displaystyle = x(x + 1)(\ln{x} - 1) + C$