Evaluate the Integral

Problem

(∫_^)(1/√(,x2−25)*d(x))

Solution

1. Identify the form of the integral as a trigonometric substitution problem involving √(,x2−a2) where a=5

2. Substitute x=5*sec(θ) which implies d(x)=5*sec(θ)*tan(θ)*d(θ)

3. Simplify the radical using the identity sec2(θ)−1=tan2(θ) resulting in √(,25*sec2(θ)−25)=5*tan(θ)

4. Rewrite the integral in terms of θ

(∫_^)((5*sec(θ)*tan(θ))/(5*tan(θ))*d(θ))

5. Simplify the integrand to (∫_^)(sec(θ)*d(θ))

6. Integrate the secant function:

ln(sec(θ)+tan(θ))+C

7. Back-substitute to return to x using sec(θ)=x/5 and tan(θ)=√(,x2−25)/5

ln(x/5+√(,x2−25)/5)+C

8. Simplify the logarithmic expression by combining terms and absorbing the constant ln(1/5) into the constant of integration C

ln(x+√(,x2−25))+C

Final Answer

(∫_^)(1/√(,x2−25)*d(x))=ln(x+√(,x2−25))+C

Want more problems? Check here!