Question

Find the indefinite integral. (Note: Solve by the simplest method—not all require integration by parts. Use C for the constant of integration.)?

∫ t ln (t+3) dt

Answer 1

To find the indefinite integral of ∫ t ln(t+3) dt, we can use integration by parts. Integration by parts is a technique used to find the integral of a product of two functions. Using the formula for integration by parts, we can find the indefinite integral of the given expression.

Step-by-step explanation:

To find the indefinite integral of ∫ t ln(t+3) dt, we can use integration by parts. Integration by parts is a technique used to find the integral of a product of two functions.

Using the formula for integration by parts: ∫ u dv = uv - ∫ v du, we can let:

u = ln(t+3) and dv = t dt.

That means du = (1/(t+3)) dt and v = (1/2) t^2.

Substituting these values into the formula, we get:

∫ t ln(t+3) dt = (1/2) t^2 ln(t+3) - ∫ (1/2) t^2 (1/(t+3)) dt.

We can simplify the integral on the right side by combining the terms and then integrate. Finally, we add the constant of integration, C, to get the result.