Integral of √(25+x²) without arcsinh

[andrew410]

I need to get the integral for sqrt(25+x^2). I can't seem to get the answer without a arcsinh. Can anyone help me? Thx in advance! :)

 

[Data]

Substitute $$x = 5\tan{\theta}$$.

You may need the identity $$\int \sec{\theta} d\theta = \ln |\sec{\theta} + \tan{\theta}| + C$$ to evaluate the resulting integral.

 

[HallsofIvy]

Generally speaking, seeing "sqrt(25+ x²)" should make you think of
a²+ b²= c² (the Pythagorean theorem) and then
(a/b)²+ 1= (c/a)² or "tan²(θ)+ 1= csc²(θ)"

Factoring a "5" out of the squareroot gives 5 sqrt(1+ (x/5)²) which leads us to think "x/5= tan(θ) or Data's x= 5 tan(θ).