The Integral of dx / √(a² – x²) | Trigonometric Substitution Method Explained

∫ dx/(sqrt(a² – x²))?

The given integral is ∫ dx / √(a² – x²)

The given integral is ∫ dx / √(a² – x²). This is a definite integral with no limits provided, so we will assume that we are finding the antiderivative or indefinite integral of this expression.

To solve this integral, let’s use a trigonometric substitution. We choose x = a sin(u), where -π/2 ≤ u ≤ π/2. Taking the derivative with respect to u, we have dx = a cos(u) du.

Now we substitute these values into the integral:

∫ dx / √(a² – x²) = ∫ (a cos(u) du) / √(a² – (a sin(u))²).

By simplifying, we get:

= ∫ (a cos(u) du) / √(a² – a²sin²(u))
= ∫ (a cos(u) du) / √(a² (1 – sin²(u)))
= ∫ (a cos(u) du) / √(a²cos²(u))
= ∫ (a cos(u) du) / (a cos(u))
= ∫ du
= u + C,
where C is the constant of integration.

Now we need to express our answer in terms of x instead of u. Using the trigonometric identity sin²(u) + cos²(u) = 1, we can solve for cos(u) as cos(u) = √(1 – sin²(u)). Applying this identity to our solution:

u + C = arcsin(x/a) + C.

Thus, the antiderivative of dx / √(a² – x²) is arcsin(x/a) + C, where C is the constant of integration.

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