1. ## trig substitutions

I have having trouble doing this one:
I set x = tan(u)
dx = sec^2(x)du
here is what i did:
integral of 2tan(u)
this equals 2sec^2(u) + C
sec(u) in this scenario is (x+2)/(2) since tan(u) = x/2
I did this then:
2((x+2/2))^2
The answer after doing this isnt the same in the book
I am suppose to get: 1/2xradical(x^2 + 4) + 2ln(x + radical(x^2 + 4)) + C

2. $\displaystyle \int\sqrt{x^{2}+4}dx$

Let $\displaystyle x=2tan(u), \;\ dx=2sec^{2}(u)du$

$\displaystyle \int\sqrt{(2tan(u))^{2}+4}\cdot{2sec^{2}(u)}du$

$\displaystyle \int\sqrt{4tan^{2}(u)+4}\cdot{2sec^{2}(u)}du$

$\displaystyle \int\sqrt{4(tan^{2}(u)+1)}\cdot{2sec^{2}(u)}du$

$\displaystyle \int(2sec(u))2sec^{2}(u)du$

$\displaystyle 4\int{sec^{3}(u)}du$

$\displaystyle \int{sec^{n}(u)}du=\frac{sec^{n-2}(u)tan(u)}{n-1}+\frac{n-2}{n-1}\int{sec^{n-2}(u)}du$

After you evaluate the integral, you can resub $\displaystyle u=tan^{-1}(\frac{x}{2})$ to get it back in terms of x.

3. Originally Posted by davecs77
I have having trouble doing this one:
I set x = tan(u)
dx = sec^2(x)du
here is what i did:
integral of 2tan(u)
this equals 2sec^2(u) + C
sec(u) in this scenario is (x+2)/(2) since tan(u) = x/2
I did this then:
2((x+2/2))^2
The answer after doing this isnt the same in the book
I am suppose to get: 1/2xradical(x^2 + 4) + 2ln(x + radical(x^2 + 4)) + C
The number in red above is an error. And I more or less can't follow what you did after that. So....
$\displaystyle \int \sqrt{x^2 + 4}dx$

Use $\displaystyle x = 2tan(u)$ --> $\displaystyle dx = 2sec^2(u) du$

So
$\displaystyle \int \sqrt{x^2 + 4}dx = \int \sqrt{4tan^2(u) + 4} \cdot 2 sec^2(u) du$

$\displaystyle = 2 \int \sqrt{4(tan^2(u) + 1)}sec^2(u) du$

$\displaystyle = 4 \int \sqrt{tan^2(u) + 1}sec^2(u) du$

$\displaystyle = 4 \int \sqrt{sec^2(u)}sec^2(u) du$

$\displaystyle = 4 \int sec(u) \cdot sec^2(u) du$

$\displaystyle = 4 \int sec^3(u) du$

Can you do it from here?

-Dan

4. Also

$\displaystyle \int\sec^3x~dx=\color{red}\int\sec{x}\sec^2x~dx$

Then you can apply integration by parts, not hard.