∫√25x2−1225dx

The integral ∫√(25x^2 - 1225) dx is a standard form of integral that can be solved using trigonometric substitution. Here are the steps to solve it:

Step 1: Identify the form of the integral The integral is in the form of ∫√(a^2 - x^2) dx, where a = √1225 = 35.

Step 2: Use trigonometric substitution For integrals of the form ∫√(a^2 - x^2) dx, we can use the substitution x = a sin θ. In this case, we substitute x = 35 sin θ, so dx = 35 cos θ dθ.

Step 3: Substitute into the integral Substituting x = 35 sin θ into the integral gives us ∫√(25 * (35 sin θ)^2 - 1225) * 35 cos θ dθ = ∫√(25 * 1225 sin^2 θ - 1225) * 35 cos θ dθ = ∫√(1225 * (25 sin^2 θ - 1)) * 35 cos θ dθ = ∫35 * √(25 sin^2 θ - 1) * 35 cos θ dθ = ∫35^2 * √(25 sin^2 θ - 1) cos θ dθ.

Step 4: Simplify the integral The integral simplifies to ∫35^2 * √(625 sin^2 θ - 25) cos θ dθ = ∫35^2 * √25 * √(25 sin^2 θ - 1) cos θ dθ = ∫35^2 * 5 * √(25 sin^2 θ - 1) cos θ dθ = ∫875 * √(25 sin^2 θ - 1) cos θ dθ.

Step 5: Solve the integral The integral ∫875 * √(25 sin^2 θ - 1) cos θ dθ can be solved using standard integral techniques. The result is 875 * ∫√(25 sin^2 θ - 1) cos θ dθ.

Step 6: Convert back to x Finally, we convert back to x using the substitution x = 35 sin θ. The final result is 875 * ∫√(25x^2/35^2 - 1) dx.

Please note that this is a complex integral and may require further simplification or the use of special functions to solve.