The Correct Answer and Explanation is:
We are asked to evaluate the integral∫3xln(3x) dx\int 3x \ln(3x) \, dx∫3xln(3x)dx
using integration by parts.
Step 1: Use substitution to simplify
Letu=3x⇒du=3dx⇒dx=13duu = 3x \Rightarrow du = 3dx \Rightarrow dx = \frac{1}{3} duu=3x⇒du=3dx⇒dx=31du
Substitute into the integral:∫3xln(3x) dx=∫uln(u)⋅13 du=13∫uln(u) du\int 3x \ln(3x) \, dx = \int u \ln(u) \cdot \frac{1}{3} \, du = \frac{1}{3} \int u \ln(u) \, du∫3xln(3x)dx=∫uln(u)⋅31du=31∫uln(u)du
Step 2: Apply integration by parts
Use the integration by parts formula:∫u dv=uv−∫v du\int u \, dv = uv – \int v \, du∫udv=uv−∫vdu
Let:
- a=ln(u)⇒da=1udua = \ln(u) \Rightarrow da = \frac{1}{u} dua=ln(u)⇒da=u1du
- db=u du⇒b=u22db = u \, du \Rightarrow b = \frac{u^2}{2}db=udu⇒b=2u2
Now apply the formula:∫uln(u) du=u22ln(u)−∫u22⋅1udu\int u \ln(u) \, du = \frac{u^2}{2} \ln(u) – \int \frac{u^2}{2} \cdot \frac{1}{u} du∫uln(u)du=2u2ln(u)−∫2u2⋅u1du=u22ln(u)−∫u2du=u22ln(u)−14u2= \frac{u^2}{2} \ln(u) – \int \frac{u}{2} du = \frac{u^2}{2} \ln(u) – \frac{1}{4} u^2=2u2ln(u)−∫2udu=2u2ln(u)−41u2
Step 3: Multiply by the constant from substitution
13(u22ln(u)−14u2)=u26ln(u)−u212\frac{1}{3} \left( \frac{u^2}{2} \ln(u) – \frac{1}{4} u^2 \right) = \frac{u^2}{6} \ln(u) – \frac{u^2}{12}31(2u2ln(u)−41u2)=6u2ln(u)−12u2
Recall that u=3xu = 3xu=3x. So:u2=(3x)2=9x2u^2 = (3x)^2 = 9x^2u2=(3x)2=9×2∫3xln(3x) dx=9x26ln(3x)−9×212+C\int 3x \ln(3x) \, dx = \frac{9x^2}{6} \ln(3x) – \frac{9x^2}{12} + C∫3xln(3x)dx=69×2ln(3x)−129×2+C
Simplify:=32x2ln(3x)−34×2+C= \frac{3}{2} x^2 \ln(3x) – \frac{3}{4} x^2 + C=23x2ln(3x)−43x2+C
Final Answer:
∫3xln(3x) dx=32x2ln(3x)−34×2+C\boxed{\int 3x \ln(3x) \, dx = \frac{3}{2} x^2 \ln(3x) – \frac{3}{4} x^2 + C}∫3xln(3x)dx=23x2ln(3x)−43x2+C
This method uses substitution followed by integration by parts. The key idea is to simplify the integral before applying the formula. This helps in correctly identifying the parts and avoiding complex expressions. The natural log function integrates well using this approach because its derivative is simpler, making it a good choice for the uuu part.
