Question

Take the integral of (x^2-7)^(3/2)dx

Please help!!!

Answer 1

\[\large \int\limits \left(x^2-7\right)^{3/2}dx\]
Ok so we have to apply a Trig-substitution.
Notice that what's in the brackets is of the form \(\large x^2-a^2\).

There are 3 types of trig subs that we can make.
When it is of this form, we want to let \(\large x=a\sec\theta\).

When we do this, here is what will happen,\[\large x^2-a^2 \qquad \rightarrow \qquad a^2\sec^2\theta-a^2 \qquad \rightarrow \qquad a^2(\sec^2\theta-1)\]
Recalling that \(\large \color{salmon}{\sec^2\theta-1=\tan^2\theta}\) gives us, \(\large a^2(\tan^2 \theta)\)

Answer 2

What this does for us is,
It simplifies the part in the brackets down to a single term.
There is no more addition/subtraction inside.
So we can now apply the power on the outside to it very conveniently.

Answer 3

So applying this idea to our problem, we'll let \(\large x=\sqrt{7}\sec\theta\).

Our integral becomes,\[\large \int\limits\limits \left(\color{orangered}{x^2}-7\right)^{3/2}dx \qquad \rightarrow \qquad \large \int\limits\limits \left(\color{orangered}{7\sec^2\theta}-7\right)^{3/2}dx\]Which will simplify to,\[\large 7^{3/2} \int\limits (\sec^2\theta-1)^{3/2}dx \qquad \rightarrow \qquad 7^{3/2}\int\limits \tan^3\theta dx\]

Answer 4

Following along ok? :D
I know these problems can be a bit tough if you're just learning trig subs.

Answer 5

Yes, I am following :)

Answer 6

So now we need to deal with the dx.
\[\large x=\sqrt{7}\sec\theta \qquad \rightarrow \qquad dx=\sqrt7\sec \theta \tan \theta \; d \theta\]

So our integral becomes,\[\large 7^{3/2}\int\limits \tan^3\theta (dx) \qquad \rightarrow \qquad \large 7^{3/2}\int\limits \tan^3\theta \left(7^{1/2}\sec \theta \tan \theta \; d \theta\right)\]Simplifying to,\[\large 49 \int\limits \tan^4\theta \sec \theta \; d \theta\]

Answer 7

Hmm lemme think a sec on this one :D
I'm sure it will jump out at me in a sec.

Answer 8

Oh this is going to be awful from this point lol...
Like.. normally once you get through the substitution, get everything plugged in, your teacher will most likely have been kind and given you something simple to integrate since you were able to get through all that.

But not in this case T.T
We once again use this idea \(\large \color{salmon}{\tan^2\theta=\sec^2\theta-1}\) to change all of our terms to secants.

From there we have to apply the Reduction Formula for Secant SEVERAL TIMES.

Answer 9

I'm not sure if I have the strength to finish this one T.T lol

Answer 10

Dont worry man I just solved it...Thanks :D

Answer 11

Oh cool c: