How proof this answer is right ? ∫ ((1)/((x^2)(sqrt(1+x^2))dx =

3 Answers
Sep 9, 2015

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Explanation:

Clear image of the equation

Sep 9, 2015

Show that the derivative of what you think is the answer is the integrand in the original problem.

Explanation:

Quick example:
To prove that #inte^(3x) dx = 1/3 e^(3x)+C#, show that the derivative of #1/3 e^(3x)+C# is #e^(3x)#:

#1/3 e^(3x)*3 = e^(3x)#. So the answer is correct.

In your question, show that the derivative of #-sqrt(1+x^2)/x +C# is #1/(x^2sqrt(1+x^2))#

It is clear that we can ignore the #+C# because the derivative of a constant is #0#.

Sep 10, 2015

You could do the actual integral and check each step along the way.

#int 1/(x^2sqrt(1+x^2))dx = ?#

Let:
#x = tantheta#
#dx = sec^2thetad theta#
#sqrt(1+x^2) = sqrt(1+tan^2theta) = sectheta#
#x^2 = tan^2theta#

This is a typical trig substitution strategy and is verified in any calculus textbook that teaches this. You can check the #sqrt(1+x^2)# substitution and you will see that it is really #sqrt(1+tan^2theta) = sqrt(sec^2theta) = sectheta#.

Therefore you get:

#=> int 1/(tan^2thetasectheta)sec^2thetad theta#

Using identities #tantheta = sintheta/costheta# and #sectheta = 1/costheta#:
#= int cos^2theta/sin^2theta*1/costhetad theta#

#= int costheta/sin^2thetad theta#

Some u-substitution can be done now as well. Let:
#u = sintheta#
#du = costhetad theta#

#=> int 1/u^2du#

#= -1/u = color(green)(-1/sintheta)#

That is our temporary answer. Finally, going back to the original substitution of #x = tantheta#:
#tantheta = sintheta/costheta = x#

#sintheta = xcostheta#

#sectheta = sqrt(1+x^2) => costheta = 1/sqrt(1+x^2)#

#sintheta = x/sqrt(1+x^2) => 1/sintheta = sqrt(1+x^2)/x#

Therefore the final answer is indeed:

#= -1/sintheta + C = color(blue)(-sqrt(1+x^2)/x + C)#