Question

Question #37d26

Answer 1

`int_0^2sqrt(4y^2-8y+5) dy=sqrt5+1/2log(sqrt5+2)~~2.957`

Explanation:

`intsqrt(x^2+a^2)=x/2sqrt(x^2+a^2)+a^2/2log (x+sqrt(x^2+a^2))`.......Property 1
`int_a^bf(x)dx=int_a^bf(a+b-x)dx` ........Property 2

`int_0^2sqrt(4y^2-8y+5) dy=2int_0^2sqrt(y^2-2y+5/4)dy`
`rArrint_0^2sqrt(4y^2-8y+5) dy=2int_0^2sqrt((y-1)^2+(1/2)^2`
Let `y-1=t`
then `dy=dt`
when `y=0 ,t=-1`
`when y=2,t=1`

`rArrint_0^2sqrt(4y^2-8y+5) dy=2int_-1^1sqrt(t^2+(1/2)^2)`
Now use property 2

`2int_-1^1sqrt(t^2+(1/2)^2)=2xx2int_0^1sqrt(t^2+(1/2)^2)`
Now use property 1
`=4xxt/2sqrt(t^2+(1/2)^2)+4xx1/2xx(1/2)^2log(t+sqrt(t^2+(1/2)^2))`
Now putting the limits we get

`2xx2int_0^1sqrt(t^2+(1/2)^2)=[2xxsqrt5/2+1/2xxlog(1+sqrt5/2)]-[1/2xxlog(1/2)]`
`=sqrt5+1/2log(sqrt5+2)`

Answer 2

`I = sqrt(5) + 1/4ln|(sqrt(5) + 2)/(sqrt(5) - 2)|`

Explanation:

Here's another method. We call the integral `I`.

Complete the square under the `√`.

`I = int_0^2 sqrt(4(y^2 - 2y) + 5) dy`

`I = int_0^2 sqrt(4(y^2 - 2y + 1 - 1) + 5) dy`

`I = int_0^2 sqrt(4(y^2 - 2y + 1) - 4 + 5) dy`

`I =int_0^2 sqrt(4(y - 1)^2 + 1) dy`

We now let `u = y - 1`. Then `du = dy`.

`I = int_(-1)^1 sqrt(4u^2 +1) du`

We now make the substitution `u = 1/2tantheta`. Then it follows that `du = 1/2sec^2theta d theta`.

`I = int_(-1)^1 sqrt(4(1/2tantheta)^2 + 1) * 1/2sec^2theta d theta`

`I=int_(-1)^1 sqrt(tan^2theta + 1) * 1/2sec^2theta d theta`

`I = int_(-1)^1 sqrt(sec^2theta) * 1/2sec^2theta d theta`

`I = int_(-1)^1 1/2sec^3theta d theta`

This is a known integral that is derived here

`I = [1/4secthetatantheta + 1/4ln|sectheta + tantheta|]_(-1)^1`

We return to `u`, where we can evaluate the integral (I didn't change the bounds of integration, so the answer that you would get from evaluating in `theta` would be wrong).

From our trig substitution, we obtain that `2u = tantheta`. So, the side opposite `theta` measures `2u` and the side adjacent measures `1`. Then the hypotenuse would measure `sqrt(4u^2 + 1)`.

This means that `sectheta = sqrt(4u^2 + 1)/1 = sqrt(4u^2 + 1)`.

`I = [1/4sqrt(4u^2 + 1)(2u) + 1/4ln|sqrt(4u^2 + 1) + 2u|]_(-1)^1`

`I = 1/2sqrt(5) + 1/4ln|sqrt(5) + 2| - (-1/2sqrt(5) + 1/4ln|sqrt(5) - 2|)`

`I = 1/2sqrt(5) + 1/4ln|sqrt(5) + 2| + 1/2sqrt(5) - 1/4ln|sqrt(5) - 2|`

`I = sqrt(5) + 1/4ln|(sqrt(5) + 2)/(sqrt(5) - 2)|`

Hopefully this helps!