Evaluate the integral

Question

∫₁⁹ (x -1)/(x^1/2) dx

 

 

 

ANSWER: 40/3

#29, p395

Answer 1

Hi Mia,

 

In order to evaluate this definite integral start by finding an anti-derivative for (x-1)/x^1/2. Dividing both terms of the numerator by the denominator (essentially re-writing the expression as two different fractions) and using rational exponents we get x/x^1/2 - 1/x^1/2 = x^1/2 - x^-1/2

 

The anti-derivative is 2/3 (x)^3/2 -1 (2) x^1/2 (raise exponent by one and multiply by reciprocal of new exponent)

 

Now "evaluate" the anti-derivative (call it F(x)) from 1 to 9. That means determine F(9) - F(1).

 

So: [(2/3 (9)^3/2 -2 (9)^1/2) - (2/3(1)^3/2 - 2(1)^1/2)]

 

= [(2/3 (27) - 2(3)) - (2/3 - 2)]

 

= [(18 - 6) - (-4/3)] = 12 +4/3 = 36/3 + 4/3 = 40/3.