Question

Find the indefinite integral of t ln(t³) dt. (Note: Solve by the simplest method–not all require integration by parts. Use c for the constant of integration.)

Answer 1

To find the indefinite integral of t ln(t³) dt, we can use integration by parts. The indefinite integral is (1/2)t² ln(t³) - (1/4)t² + c, where c is the constant of integration.

Step-by-step explanation:

To find the indefinite integral of t ln(t³) dt, we can use integration by parts. Let u = ln(t³) and dv = t dt. Taking the derivative of u, we get du = (1/t) dt. Integrating dv, we get v = (1/2)t². Now, we can use the integration by parts formula:

∫ udv = uv - ∫ vdu

Substituting the values, we get:

∫ t ln(t³) dt = (1/2)t² ln(t³) - (1/2) ∫ t²(1/t) dt

Simplifying further, we have:

∫ t ln(t³) dt = (1/2)t² ln(t³) - (1/2) ∫ t dt

Integrating the second term, we get:

∫ t ln(t³) dt = (1/2)t² ln(t³) - (1/4)t² + c

Therefore, the indefinite integral of t ln(t³) dt is (1/2)t² ln(t³) - (1/4)t² + c, where c is the constant of integration.