(x^3-1)^1/3 X^5 Integral

3 min read Jun 17, 2024

Evaluating the Integral of (x^3-1)^(1/3) x^5

This article explores the process of evaluating the definite integral of (x^3-1)^(1/3) x^5. We will use a combination of substitution and integration by parts to solve this problem.

1. Substitution

Let's start by simplifying the integrand using substitution.

Let u = x^3 - 1. Then, du = 3x^2 dx. We can rewrite the integral as follows:

(x^3 - 1)^(1/3) x^5 dx = (u)^(1/3) * (1/3) * (u + 1) du

Note: We replaced x^5 with (1/3)*(u+1) since x^3 = u + 1.

2. Integration by Parts

Now, we can apply integration by parts.

Let dv = (u)^(1/3) * (u + 1) du. Then, v = (3/4) * u^(4/3) * (u + 1) - (9/28) * u^(7/3).

Let u = 1 (a constant). Then, du = 0.

Applying integration by parts formula:

∫ (x^3-1)^(1/3) x^5 dx = ∫ (u)^(1/3) * (u + 1) du = uv - ∫ v du

= (1) * [(3/4) * u^(4/3) * (u + 1) - (9/28) * u^(7/3)] - ∫ [(3/4) * u^(4/3) * (u + 1) - (9/28) * u^(7/3)] * 0 du

= (3/4) * u^(4/3) * (u + 1) - (9/28) * u^(7/3)

3. Back Substitution

Finally, we substitute u = x^3 - 1 back into the result:

(3/4) * u^(4/3) * (u + 1) - (9/28) * u^(7/3) = (3/4) * (x^3 - 1)^(4/3) * (x^3) - (9/28) * (x^3 - 1)^(7/3)

4. The Final Solution

Therefore, the integral of (x^3-1)^(1/3) x^5 is:

(3/4) * (x^3 - 1)^(4/3) * (x^3) - (9/28) * (x^3 - 1)^(7/3) + C

Where C is the constant of integration.