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

[bawd]

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]

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

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)

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]

No guarantees of getting the last one correct. =P