Trig substitution for integral of x²/√(4x²+8)

[ProBasket]

i need to find the appropriate trigonometric substitution for this problem. i don't know why, but I am only having problems finding the right sub. i can do the rest pretty easily. please help me with this:

$$\int \frac{x^2}{sqrt(4x^2+8)}$$

here's what i done:

sqrt(4(x^2+2)) ----> a*tan(theta)

so I'm thinking that x should be equal to sqrt(2)/2*tan(theta), but it's incorrect. can someone help? (i know it's the wrong sub because I'm doing my homework online and it askes me for the trig sub first to check)

 

[Justin Lazear]

Why 1/sqrt(2) and not just sqrt(2)?

--J

 

[dextercioby]

How about a substitutiton involving $\sinh$ ...?After all,it's still a trigonometric function,except,that it's not circular.

Daniel.

P.S.It should give the result immediately.

 

[Kelvin]

i need to find the appropriate trigonometric substitution for this problem. i don't know why, but I am only having problems finding the right sub. i can do the rest pretty easily. please help me with this:

$$\int \frac{x^2}{sqrt(4x^2+8)}$$

here's what i done:

sqrt(4(x^2+2)) ----> a*tan(theta)

so I'm thinking that x should be equal to sqrt(2)/2*tan(theta), but it's incorrect. can someone help? (i know it's the wrong sub because I'm doing my homework online and it askes me for the trig sub first to check)

can i try?

$$\[<br /> \int_{}^{} {\frac{{x^2 }}{{\sqrt {4x^2 + 8} }}} dx \\ \\<br /> = \int_{}^{} {\frac{{x^2 }}{{2\sqrt {x^2 + 2} }}} dx \\$$
let
$$x = \sqrt 2 \tan \theta$$

$$\[<br /> \int_{}^{} {\frac{{2\tan ^2 \theta \sqrt 2 \sec ^2 \theta }}{{2\sqrt 2 \sec \theta }}d\theta } \\ <br /> = \int_{}^{} {\tan ^2 \theta } \sec \theta d\theta \\ <br /> = \int_{}^{} {\sec \theta d \sec \theta } \\ <br /> \]<br />$$

 

[Gamma]

you know that

$$1 + tan^2\theta = sec^2 \theta$$

Use the subtitution, $$x = \sqrt{2} tan \theta$$

 

[Gamma]

Oh I see when I finished typing the latex, somebody posted the solution.

 

[ProBasket]

ah thanks for the help. don't know why i added the 1/2.

we haven't gone over sinh yet and don't think we ever will, so it'll be better if i don't use it.