How to integrate int (x^2 + 4)^(-1/2) dx Is it a substitution?

[gordda]

I need help antidiffing this equation:
(x^2+4)^(-1/2)

i have tried subbing u=x^2+4 i have tried subbing u= (x^2+4)^(-1/2).
i have tried making x the subject. even tried to use partial fraction, with no avail, because i could not figure out how to use partially factorize it.
If anyone could lead me towards the right answer.
Thanks.

 

[saltydog]

I need help antidiffing this equation:
(x^2+4)^(-1/2)

i have tried subbing u=x^2+4 i have tried subbing u= (x^2+4)^(-1/2).
i have tried making x the subject. even tried to use partial fraction, with no avail, because i could not figure out how to use partially factorize it.
If anyone could lead me towards the right answer.
Thanks.

Whenever you have an integrand that looks like the Pathagoras theorem, use trig substitutions. You know, draw a right triangle with theta in it. Then the hypotneuse is the radical, one side is 2, the other is x right. Follow through: What would the tan(theta) be?

 

[thepatientmental]

Yes, integrating (x^2+4)^(-1/2) can be done using substitution. Here's how:

Let u = x^2 + 4
Then du/dx = 2x
And dx = du/2x

Substituting these values into the original equation, we get:

∫ (x^2+4)^(-1/2) dx = ∫ (x^2+4)^(-1/2) * (du/2x)
= (1/2) * ∫ u^(-1/2) du
= (1/2) * ∫ u^(-1/2) * u^0 du
= (1/2) * ∫ u^(1/2) du
= (1/2) * (2u^(3/2)) + C
= u^(3/2) + C
= (x^2+4)^(3/2) + C

So, the final integrated form is (x^2+4)^(3/2) + C. You can also verify this solution by differentiating it and checking if it gives the original equation.

Hope this helps!