How do you integrate #intdx/ sqrt(x^2 - a^2)#?

Answer 1

#int dx/sqrt(x^2-a^2) = ln abs (x +sqrt(x^2-a^2)) +C#

The integrand is defined for #x in (-oo,-a) uu (a,+oo)#. Let us focus first on #x in (a,+oo)# and substitute:
#x = a sect#
#dx = asect tant#
with #t in (0,pi/2)#

so:

#int dx/sqrt(x^2-a^2) = int (asect tant dt)/sqrt(a^2sec^2t-a^2)#
#int dx/sqrt(x^2-a^2) = int (sect tant dt)/sqrt(sec^2t-1)#

Use now the trigonometric identity:

#sec^2t-1 = tan^2t#
and as for #t in (0,pi/2)# the tangent is positive:
#sqrt(sec^2t-1) = tan^#

Then:

#int dx/sqrt(x^2-a^2) = int (sect tant dt)/tant#
#int dx/sqrt(x^2-a^2) = int sect dt#
#int dx/sqrt(x^2-a^2) = ln abs (sect +tant) +C#

Undoing the substitution:

#int dx/sqrt(x^2-a^2) = ln abs (x/a +sqrt(x^2/a^2-1)) +C#
#int dx/sqrt(x^2-a^2) = ln abs (x +sqrt(x^2-a^2)) +C#
By differentiating we can see that the solution is valid also for #x in (-oo,-3)#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

#intdx/sqrt(x^2-a^2)="arcosh"(x/a)+"c"#

#intdx/sqrt(x^2-a^2)=intdx/(asqrt(x^2/a^2-1))=int1/(asqrt((x/a)^2-1))dx#
Now, let #u=x/a# and #du=1/adx#

Then

#int1/(asqrt((x/a)^2-1))dx=int1/sqrt(u^2-1)dx="arcosh"(u)+"c"="arcosh"(x/a)+"c"#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To integrate ( \int \frac{dx}{\sqrt{x^2 - a^2}} ), where ( a ) is a constant:

  1. Make a trigonometric substitution ( x = a \sec(\theta) ).
  2. Compute ( dx = a \sec(\theta) \tan(\theta) d\theta ).
  3. Substitute ( x ) and ( dx ) in terms of ( \theta ) into the integral.
  4. Simplify the integral in terms of ( \theta ).
  5. Integrate the simplified expression with respect to ( \theta ).
  6. Substitute back ( \theta ) in terms of ( x ) to get the final answer.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7