How do you prove the integral formula #intdx/(sqrt(x^2+a^2)) = ln(x+sqrt(x^2+a^2))+ C# ?

Answer 1
Let #x=atan theta#. #Rightarrow dx=asec^2theta d theta# So, we can write #int{dx}/{sqrt{x^2+a^2}}=int{asec^2theta}/{sqrt{a^2(tan^2theta+1)}}d theta# by #tan^2theta+1=sec^2theta#, #=intsec theta d theta=ln|sec theta+tan theta|+C# since #tan theta=x/a# and #sec theta={sqrt{x^2+a^2}}/a#, #=ln|{sqrt{x^2+a^2}+x}/a|+C_1# by the log property #ln{A/B}=lnA-lnB#, #=ln|sqrt{x^2+a^2}+x|-ln|a|+C_1# by setting #C=ln|a|+C_1#, #=ln|x+sqrt{x^2+a^2}|+C#
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Answer 2

To prove the integral formula

[ \int \frac{dx}{\sqrt{x^2 + a^2}} = \ln(x + \sqrt{x^2 + a^2}) + C ]

where ( C ) is the constant of integration, you can use a trigonometric substitution. Let ( x = a \sinh(t) ). Then, ( dx = a \cosh(t) , dt ).

Substituting these into the integral:

[ \int \frac{a \cosh(t) , dt}{\sqrt{(a\sinh(t))^2 + a^2}} ]

Simplify:

[ \int \frac{a \cosh(t) , dt}{\sqrt{a^2(\sinh(t))^2 + a^2}} ]

[ = \int \frac{a \cosh(t) , dt}{\sqrt{a^2(\sinh(t))^2 + a^2}} ]

[ = \int \frac{a \cosh(t) , dt}{\sqrt{a^2(\sinh(t))^2 + a^2}} ]

[ = \int \frac{a \cosh(t) , dt}{a\sqrt{(\sinh(t))^2 + 1}} ]

[ = \int \frac{\cosh(t) , dt}{\sqrt{(\sinh(t))^2 + 1}} ]

Now, note that ( \cosh^2(t) - \sinh^2(t) = 1 ), so ( \sinh^2(t) = \cosh^2(t) - 1 ).

[ = \int \frac{\cosh(t) , dt}{\sqrt{\cosh^2(t) - 1 + 1}} ]

[ = \int \frac{\cosh(t) , dt}{\sqrt{\cosh^2(t)}} ]

[ = \int dt ]

[ = t + C ]

Finally, substitute back ( t ) in terms of ( x ):

[ t = \text{arsinh}\left(\frac{x}{a}\right) ]

[ = \text{arsinh}\left(\frac{x}{a}\right) + C ]

[ = \ln\left(x + \sqrt{x^2 + a^2}\right) + C ]

Thus, the formula is proven.

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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.

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