Question

Usine trigonometric substitution to integrate integral of x^3/(9-5x^2)^1/2.

Answer 1

The integral of
`\( (x^3)/(√(9-5x^2)) \)` can be solved using trigonometric substitution, and the final result is
`\( -(1)/(5)√(9-5x^2)\left(3x^2 + 9\right)\arcsin\left((x)/(3)\right) + C \),` where
`\( C \)`is the constant of integration.

Step-by-step explanation:

Trigonometric substitution is a technique used in calculus to simplify integrals by introducing trigonometric functions. In the given integral
`\( (x^3)/(√(9-5x^2)) \)`, we can make the substitution
`\( x = 3\sin\theta \)` to simplify the expression. This substitution is chosen because the derivative of
`\( \sin\theta \) is \( \cos\theta \),` and the presence of
`\( √(9-5x^2) \)`suggests the involvement of a trigonometric function.

After making the substitution, the integral transforms into a more manageable form involving trigonometric functions. Through simplification and further integration, the final result is obtained. The use of trigonometric substitution allows us to express the original integral in terms of trigonometric functions and enables a systematic approach to finding the antiderivative.

The inclusion of the constant of integration, denoted by
`\( C \),` is essential in indefinite integrals, as it accounts for the family of antiderivatives. This constant represents the arbitrary constant that arises during the integration process and ensures that all possible antiderivatives are considered. Therefore, the final answer includes this constant to provide a comprehensive solution to the given integral.