Final answer:
The integral ∫ (dx)/(x√8x+25) can be evaluated by substituting u = 8x + 25, leading to a simpler integral in terms of u. After finding the antiderivative, the final solution is (1/4)√(8x + 25) + C.
Step-by-step explanation:
To evaluate the integral ∫ (dx)/(x√8x+25), a common technique is substitution. Since the denominator involves a square root, one might look for a substitution that simplifies the square root expression. Let's consider the expression inside the root, 8x + 25. We can perform a substitution such as u = 8x + 25, which would then allow us to rewrite the integral in terms of u.
First, we differentiate u to find du: du = 8dx. We can then solve for dx to substitute back into our integral: dx = du/8. Now, we rewrite the integral in terms of u: ∫ (1/8)du/(u^(1/2)), with the x in the denominator replaced by (u - 25)/8.
We can now integrate with respect to u to find the antiderivative: ∫ (1/8) u^(-1/2) du. The result of this integral is (1/8) * 2u^(1/2) + C, where C is the constant of integration. Substituting back for u, we get (1/4)√(8x + 25) + C as the final answer.