Solve for Integral of √(25+x²) with Help | Calc Problem Assistance

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To solve the integral of √(25+x²), the substitution x = 5tan(θ) is recommended. This transformation simplifies the expression, allowing the use of the identity ∫sec(θ) dθ = ln|sec(θ) + tan(θ)| + C for evaluation. The problem relates to the Pythagorean theorem, suggesting a connection between the integral and trigonometric identities. Factoring out a 5 from the square root leads to the expression 5√(1 + (x/5)²), reinforcing the substitution. This approach provides a clear pathway to finding the integral without needing arcsinh.
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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! :)
 
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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.
 
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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).
 

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