The integral ∫t ln(t + 2) dt can be solved using integration by parts, applying the formula ∫udv = uv - ∫vdu. The correct assignments are u = ln(t + 2) and dv = t dt, leading to v = t²/2. The solution involves simplifying the integral into manageable parts, specifically ∫(t²/(t + 2)) dt, which can be further broken down into simpler integrals. The final result includes a constant of integration, C, and requires careful attention to each step to avoid errors.
PREREQUISITESStudents studying calculus, particularly those focusing on integration techniques, as well as educators looking for examples of integration by parts in practice.
Leave that red part out.chapsticks said:Homework Statement
Find the integral. (Note: Solve by the simplest method - not all require integration by parts.)
∫t ln(t + 2) dt
Homework Equations
∫udv=uv-∫vdu
The Attempt at a Solution
dv=tdt v=tdt=t2/2
u=ln(t+2) du=1/(t+2)dt
Check the vdu in that last integral.∫t ln(t + 2) dt
=(t2/2)ln(t+2)-(1/2)∫(t2/2)dt
chapsticks said:Homework Statement
Find the integral. (Note: Solve by the simplest method - not all require integration by parts.)
∫t ln(t + 2) dt
Homework Equations
∫udv=uv-∫vdu
The Attempt at a Solution
dv=tdt v=tdt=t2/2
u=ln(t+2) du=1/(t+2)dt
∫t ln(t + 2) dt
=(t2/2)ln(t+2)-(1/2)∫(t2/2)dt
=(t2/2)ln(t+2)-(1/2)∫(t-2+(4/(t+2))dt
=(t2/2)ln(t+2)-(1/2)[(t2/2)-2t+4ln(t+2)]+C
=? Idk why I keep getting it wrong?