A few random questions: log, U-sub, area, trig (1 Viewer)

bawd · Feb 19, 2009

Just some questions I from some random worksheets I couldn't do. Please show some basic working out as well. Thanks. =)

1. Evaluate x√(x^2 -4) dx upper limit: 3, lower limit: 2 using the substitution u = x^2 - 4

2. Calculate the area of the region bounded by the curve y = x / √(x ^2 + 4), the x-axis and the lines x = 0 and x = 2√3 . Use the substitution u = x ^2 +4

3. Write an expression for tan (a - b ) and use your result to show that tan(times)pi/12 = (√3 - 1)/(√3 + 1)

4. Given y = (4x + 3)e^(-2x), show that y" + 4y' + 4y = 0

Thanks again.

tommykins · Feb 19, 2009

bawd said:
1. Evaluate x√(x^2 -4) dx upper limit: 3, lower limit: 2 using the substitution u = x^2 - 4

$\int_ {2}^{3} x.\sqrt{x^{2}-4} \text{ dx}\\ \text{Let u = } x^{2} - 4\\ x = 2, u = 0\\ x = 3, u = 5\\ \frac{du}{dx} = 2x \Rightarrow dx = \frac{du}{2x}\\\\ \int_ {0}^{5} \frac{x.u^{\frac{1}{2}}}{2x} \text{ du}\\ = \frac{1}{2} \int_ {0}^{5} u^{\frac{1}{2}} du\\\\ \text{Then it becomes a normal integral.}$

2. Calculate the area of the region bounded by the curve y = x / √(x ^2 + 4), the x-axis and the lines x = 0 and x = 2√3 . Use the substitution u = x ^2 +4

Same theory as above.

3. Write an expression for tan (a - b ) and use your result to show that tan(times)pi/12 = (√3 - 1)/(√3 + 1)

$\tan (a-b) = \frac {\tan a - \tan b}{1 + \tan a\tan b}\\ \tan (\frac{\pi}{12}) = \tan (\frac{\pi}{4} - \frac{\pi}{6})\\ \text{Sub the values in and use exact values, simplify it down etc. }$

4. Given y = (4x + 3)e^(-2x), show that y" + 4y' + 4y = 0

Thanks again.

Simply expand and differentiate and collect y, y' and y''.
Then simplify and it'll equate to 0.

Aerath · Feb 19, 2009

No guarantees of getting the last one correct. =P

A few random questions: log, U-sub, area, trig (1 Viewer)

bawd

Member

tommykins

i am number -e^i*pi

Aerath

Retired

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