Question

12th maths ∫1/√4x²-9

Answer 1

Answer:

Sorry this question is very hard

Answer 2

Explanation:

(x−a)(b−x)=bx−x2−ab+ax=−x2+(a+b)x−ab

=−x2+(a+b)x+(a+b)2−4−(a+b)2−4−ab

=−(x−a+b2)2−a2+2ab+b2−4ab4

=(a−b2)2−(x−a+b2)2.

So our integral can be written as

∫1(a−b2)2−(x−a+b2)2−−−−−−−−−−−−−−−−−√dx.

Now we use the substitution

x=a+b2+a−b2sinθ.

(Note the right hand side is not a constant since we are given that a≠b.)

dxdθ=a−b2cosθ

so that

dx=a−b2cosθdθ.

Substituting in our integral, we obtain

∫a−b2cosθ(a−b2)2−(a−b2sinθ)2−−−−−−−−−−−−−−−−−√dθ

=∫1dθ=θ+K.

Since x=a+b2+a−b2sinθ,

x−a+b2=a−b2sinθ

2a−b(x−a+b2)=sinθ

2x−a−ba−b=sinθ

θ=arcsin(2x−a−ba−b).

So finally our integral is equal to

arcsin(2x−a−ba−b)+K.