Final answer:
To solve the given integral, you can start by simplifying the expression under the square root. Then, use the substitution method to rewrite the integral in terms of a new variable. Simplify and solve the integral with respect to the new variable, and then substitute the original variable back in to obtain the final answer.
Step-by-step explanation:
To solve the integral, we can start by simplifying the expression under the square root:
√(2)√(x²-a²) = √(2(x²-a²)) = √(2x²-2a²)
Now, we can rewrite the integral as:
∫ₐ ᵃ (√(2x²-2a²))/x³ dx
To solve this integral, we can use the substitution method. Let z = x²-a². Then, dz = 2x dx.
Substituting these values, the integral becomes:
∫ₐ ᵃ (√(z)/x³) * (1/2x) dz
Now, we can simplify and solve the integral with respect to z:
∫ₐ ᵃ (√(z)/2x⁴) dz
After evaluating the integral, we can substitute z back in using the reverse substitution and simplify the final answer.