Final answer:
The indefinite integral of x³ ln(x) dx using integration by parts results in (ln(x) · x⁴/4) - x⁵/80 + C, where C is the constant of integration.
Step-by-step explanation:
To solve the indefinite integral of x³ ln(x) dx using integration by parts, we first identify the correct parts for the formula. The formula for integration by parts is ∑ udv = uv - ∑ vdu. To apply this formula, we let u = ln(x) which gives us du = dx/x, and let dv = x³ dx which yields v = x⁴/4. Now we can rewrite the integral in terms of u and v.
Substituting our chosen parts into the formula, we get:
The product of u and v: uv = (ln(x) · x⁴/4)
The integral of v du: ∑ vdu = ∑ (x⁴/4) · (dx/x) = ∑ x³/4 dx = x⁴/16
The expression for the integral we are calculating then becomes:
∑ x³ ln(x) dx = ln(x) · x⁴/4 - ∑ x⁴/16 dx
Integrating x⁴/16, we get x⁵/80. Hence, the final result is (ln(x) · x⁴/4) - x⁵/80 + C, where C is the constant of integration.