1/((2tanΦ)^2√(4+x^2))

Question

i am a high school student

Comments

Can you explain in more detail what you are looking for? The expression contains two variables but no indication about what to do with it. My guess is that it’s something to do with integration, because to integrate the square root, the technique would be to substitute x=2tanɸ.

Answer 1

I am going to make the bold assumption that you are trying to work out  ∫(dx/(x²√(4+x²))).

One way to do this is by substitution: x=2tanɸ, so dx=2sec²ɸdɸ.

Now we substitute into the integral:

∫(2sec²ɸdɸ/(4tan²ɸ√(4+4tan²ɸ)))=

∫(2sec²ɸdɸ/(4tan²ɸ(2secɸ)))= [because 1+tan²ɸ ≡ sec²ɸ as a trig identity]

∫(secɸdɸ/4tan²ɸ)=¼ʃ(cosɸdɸ/sin²ɸ).

Substitute again y=sinɸ, then dy=cosɸdɸ, so we have ¼ʃ(dy/y²)=-1/(4y)+C where C is the integration constant, because 1/y² is the same as y⁻² which integrates to -y⁻¹ = 1/y.

Now we go back through the substitutions:

-1/(4y)+C=-¼cosecɸ+C=-¼√(4+x²)/x+C=-√(4+x²)/(4x)+C.