The <em>complete</em> solution of the integral ∫ x · √(8 · x - x²) dx is equal to [- √(8 · x - x²)³ / 3] - (- 2 · x + 8) · √(8 · x - x²) + 32 · [ - sin⁻¹ [(- 2 · x + 8) / 8]].
<h3>How to find the integral of a function by table of integrals</h3>
Some integrals cannot be resolved immediately or at least in few steps, requiring sometimes the resolution of some <em>intermediate</em> integrals. In this case, we have an integral of the form:
∫ x · √(a · x² + b · x + c) dx, where a, b, c are <em>real</em> constants. (1)
The use of integral tables allows us to save time on resolution of integrals.
The solution of (1) can be found by using part integration several times. In accordance with the table of integrals, we find the following information:
∫ x · √(a · x² + b · x + c) dx = [√(a · x² + b · x + c)³ / (3 · a)] - [[b · (2 · a · x + b)] / (8 · a²)] · √(a · x² + b · x + c) - [[b · (4 · a · c - b²)] / (16 · a²)] ∫ [1 / √(a · x² + b · x + c)] dx
∫ [1 / √(a · x² + b · x + c)] dx = - (1 / √- a) · sin⁻¹ [(2 · a · x + b) / [√(b² - 4 · a · c)]]
Then, the <em>complete</em> solution of the integral is:
∫ x · √(a · x² + b · x + c) dx = [√(a · x² + b · x + c)³ / (3 · a)] - [[b · (2 · a · x + b)] / (8 · a²)] · √(a · x² + b · x + c) - [[b · (4 · a · c - b²)] / (16 · a²)] · [ - (1 / √- a) · sin⁻¹ [(2 · a · x + b) / [√(b² - 4 · a · c)]]]
If we know that a = - 1, b = 8 and c = 0, then the complete solution of the integral is:
∫ x · √(8 · x - x²) dx = [- √(8 · x - x²)³ / 3] - (- 2 · x + 8) · √(8 · x - x²) + 32 · [ - sin⁻¹ [(- 2 · x + 8) / 8]]
To learn more on integral tables: brainly.com/question/14406733
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