Question

Evaluate the following integral. (dx) / √(25-4x²)

a) sin^(-1)(2x/5) + C
b) cos^(-1)(2x/5) + C
c) tan^(-1)(2x/5) + C
d) cot^(-1)(2x/5) + C

Answer 1

To evaluate the given integral, we can use the trigonometric substitution method. The answer is cos^(-1)(2x/5) + C.

Step-by-step explanation:

The question involves evaluating an indefinite integral in the form ∞ dx / √(25 - 4x²). This is a standard integral that can be solved using a trigonometric substitution. To evaluate the integral ∫(dx)/√(25-4x²), we can use the trigonometric substitution method. Let x = (5/2)sinθ. Then dx = (5/2)cosθdθ. Substituting these values into the integral, we get:

∫[(5/2)cosθdθ]/√(25-4(25/4)sin²θ)

Simplifying further, we obtain:

(5/2)∫cosθ/√(1-sin²θ)dθ

Recognizing that this is the form of the derivative of the arccosine function (cos^(-1)(x)), the answer is:

cos^(-1)(2x/5) + C