Question

match the integral with the appropriate substitution...

Answer 1

\[\int\limits \frac{ x^5 }{ \sqrt{x^2+4} } dx\]
\[\int\limits x^3 \sqrt{4-x^2}dx\]
\[\int\limits \frac{ x^2 }{ \sqrt{x^2-4} }dx\]
\[\int\limits x \sqrt{x^2+4}dx\]

Answer 2

u = x^2 + 4
x = 2sinθ
x= 2secθ
x= 2tanθ

Answer 3

i need to match them with the integrals

Answer 4

Any ideas which one is the u-substitution? :)

Answer 5

absolutely no idea!! :(

Answer 6

@zepdrix

Answer 7

Well \(\large\rm u=x^2+4\) corresponds to the `first` or `fourth` option as those have an x^2+4 showing up.

Your du would be \(\large\rm du=2x~dx\)
or written another way: \(\large\rm \frac12du=x~dx\)

So do we have an x to the first power in either of those options?

Answer 8

Option 1:\[\large\rm \int\limits\limits \frac{ x^5 }{ \sqrt{x^2+4} } dx=\int\limits\frac{1}{\sqrt{x^2=4}}(x^5dx)\]Option 4:\[\large\rm \int\limits x \sqrt{x^2+4}~dx=\int\limits \sqrt{x^2+4}~(x~dx)\]

Answer 9

the second one

Answer 10

option 4

Answer 11

so for the others, do i have to find the derivatives of x ?

Answer 12

K good, found our u-sub.

Answer 13

No, not really.
You just need to match up the form of the stuff under the root to the `correct Pythagorean Identity`.

Here are the identities to recall:\[\large\rm \sin^2\theta+\cos^2\theta=1\qquad\qquad\to\qquad\qquad \color{royalblue}{\cos^2\theta=1-\sin^2\theta}\]\[\large\rm \color{orangered}{\tan^2\theta+1=\sec^2\theta}\qquad\qquad\to\qquad\qquad \color{green}{\tan^2\theta=\sec^2\theta-1}\]These three colored things are what we use to identify the correct form for our stuff.

Answer 14

So when we see something like: \(\large\rm x^2-4\),
We should think, that looks an awful lot like \(\large\rm x^2-1\)
And if we replaced x by secant, \(\large\rm sec^2\theta-1\)
then we would be able to use the green identity to reduce it down to \(\large\rm tan^2\theta\)

Answer 15

So I should be careful though,
we don't have -1, we have minus 4,
so we need to produce a 4 with the secant as well,
so we can factor the 4's out.

Answer 16

\(\large\rm x=2sec\theta\)
then
\(\large\rm (x)^2-4=(2sec\theta)^2-4=4sec^2\theta-4=4(sec^2\theta-1)=4(\tan^2\theta)\)

Answer 17

so x=2tanθ is matched with integral 3?

Answer 18

i mean x=2secθ

Answer 19

The third option has an \(\large\rm x^2-4\),
so yes good

Answer 20

the first one would be x=2tanθ and the second would be x=2sinθ ?

Answer 21

The identity involving addition is tangent,
good good good.
Good.

Answer 22

thanks!