Evaluate the Integral

Problem

(∫_^)(1/√(,x^2+100)*d(x))

Solution

1. Identify the form of the integral as (∫_^)(1/√(,x^2+a^2)*d(x)) where a=10

2. Apply the trigonometric substitution x=10*tan(θ) which implies d(x)=10*sec^2(θ)*d(θ)

3. Substitute these into the integral to get (∫_^)((10*sec^2(θ))/√(,100*tan^2(θ)+100)*d(θ))

4. Simplify the denominator using the identity 100*(tan^2(θ)+1)=100*sec^2(θ) resulting in (∫_^)((10*sec^2(θ))/(10*sec(θ))*d(θ))

5. Reduce the expression to (∫_^)(sec(θ)*d(θ))

6. Integrate using the standard formula ln(sec(θ)+tan(θ))+C

7. Back-substitute using tan(θ)=x/10 and sec(θ)=√(,x^2+100)/10 to return to the variable x

8. Simplify the logarithmic expression ln((√(,x^2+100)+x)/10)+C by absorbing the constant −ln(10) into the constant of integration C

Final Answer

(∫_^)(1/√(,x^2+100)*d(x))=ln(x+√(,x^2+100))+C

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