## 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.