Two integrals

[MathsFractals]

Greetings to all!


I don't know how to solve these two integrals.

1. [MATH]\int\sqrt{x^2-a^2} [/MATH] dx ......................................edited

2. [MATH]\int\sqrt{x^2+a^2}[/MATH]dx.........................................edited


For the FIRST INTEGRAL, I found on the internet that it can be solved with:

[MATH]x=a cosh t[/MATH].


For the SECOND INTEGRAL, I found on the internet that it can be solved with:

[MATH]x=a sinh t[/MATH].


However, hyperbolic functions are not taught in high school. If someone knows how to solve them in another way, it would mean a lot to me.



P. S.

I apologize for not knowing English. I hope you can understand what I am asking.

 

[Deleted member 4993]

Greetings to all!


I don't know how to solve these two integrals.

1. [MATH]\int\sqrt{x^2-a^2} dx[/MATH] ......................................edited

2. [MATH]\int\sqrt{x^2+a^2}dx[/MATH].........................................edited


For the FIRST INTEGRAL, I found on the internet that it can be solved with:

[MATH]x=a cosh t[/MATH].


For the SECOND INTEGRAL, I found on the internet that it can be solved with:

[MATH]x=a sinh t[/MATH].


However, hyperbolic functions are not taught in high school. If someone knows how to solve them in another way, it would mean a lot to me.



P. S.

I apologize for not knowing English. I hope you can understand what I am asking.

You do NOT have to use hyperbolic function.

For the first integral: substitute x = a * sec(u)

For the second integral: substitute x = a * tan(u)

Please share your work/thoughts about this problem.

 

[Steven G]

Draw a right triangle where one of the legs is sqrt(x^2-a^2). The square root of the positive term under the square root sign (this is x^2) goes on the hypotenuse and the square root of the term being subtracted (in this case a^2) goes on the other leg. Not come up with with a trig function that doe NOT involve the side with the square root.

Prof Khan's method works but requires you to remember the correct substitution.

Go to this link, where I show how to do this without memorizing anything.