What's the derivative of 1+u2 ?Rosebud said:
The derivative of 1+u^2 with respect to u is 2u du.SammyS said:
Do you know the method of substitution ?Rosebud said:
No.SammyS said:
That should have been a BIG hint.Rosebud said:
Thanks for the tip. I did that but I still fail to see the connection. Can you, or someone else, give me the next step?SammyS said:
Numerator? There is no fraction here unless you you consider 1 as the denominator.DeldotB said:
ahh, thought I saw a / in front of the root.Rosebud said:
That helped immensely. Thank you so much. I forgot that I could use substitution more than once.Rosebud said:
SammyS said:
So, you do know substitution.Rosebud said:
Rosebud said:
Yes, I know substitution. You asked me if I knew the method of substitution, which I understood as, you asking if I knew which method of substitution that I should use.SammyS said:
The integration of u√(1+u^2) du is equal to (1/3)(1+u^2)^(3/2) + C.
To solve this type of integration problem, you can use the substitution method. Let u = 1+u^2 and du = 2u du. Then, the integral can be rewritten as (1/2)∫u^(1/2) du, which can be easily integrated.
Yes, this integral can also be solved using integration by parts. Let u = u and dv = √(1+u^2) du. Then, du = du and v = (1/3)(1+u^2)^(3/2). This method may be more complicated but can also lead to the same answer.
Yes, if the integration limits are from 0 to a, where a is a positive real number, the integral can be simplified to (1/3)(a√(1+a^2) + arcsinh(a)).
No, this type of integral requires either the substitution method or integration by parts to be solved. Other methods such as partial fractions or trigonometric substitutions may not be applicable.