Question

Evaluate the indefinite integral
x⁸ / (81-x²)¹1/2

Answer 1

To evaluate the indefinite integral x⁸ / (81-x²)¹1/2, use the substitution u = 81 - x² and solve the integral step by step.

Step-by-step explanation:

To evaluate the indefinite integral ∫x⁰ / (81 - x²)¹⁄⁰, we can use the substitution u = 81 - x². This transforms the integral into ∫(81 - u)¹⁄⁰du. Let's proceed with the substitution and solve the integral step by step:

1. Substitute u = 81 - x²:

∫(81 - x²)¹⁄⁰ dx = ∫u¹⁄⁰ dx

1. Calculate du/dx and solve for dx:

du/dx = -2x, dx = -½ du/x

1. Substitute the new variables and simplify the integral:

∫(81 - u)¹⁄⁰ du = ∫(81 - u)¹⁄⁰(-½ du/x)

= -½∫(81 - u)¹⁄⁰ du

1. Integrate with respect to u:

= -½(u/2) + C

1. Undo the substitution:

= -½((81 - x²)/2) + C

= -½(6561 - 162x² + x⁰/2) + C