How can I evaluate the integral of xln(x)dx by using integration by parts
u = lnx
du = 1/x dx
dv = x dx
v = 1/2 x^2
∫u dv = uv - ∫v du, so
∫x lnx dx = 1/2 x^2 lnx - ∫(1/2 x^2)(1/x dx)
= 1/2 x^2 lnx - 1/2 ∫x dx
...
To evaluate the integral ∫xln(x)dx using integration by parts, you need to follow the steps of the technique. Integration by parts uses a specific integration formula:
∫u dv = uv - ∫v du.
Here's how you can apply integration by parts to evaluate the given integral:
Step 1: Determine u and dv:
Let u = ln(x) and dv = xdx. We choose u as ln(x) because it will simplify when differentiated.
Step 2: Calculate du and v:
Differentiate u to find du:
du = (1/x)dx
Integrate dv to find v:
∫x dx = (1/2)x^2
Step 3: Apply the integration by parts formula:
Using the integration by parts formula:
∫u dv = uv - ∫v du,
we can rewrite the given integral as:
∫x ln(x) dx = ln(x) * (1/2) * x^2 - ∫ (1/2) * x^2 * (1/x) dx
Simplifying further:
∫x ln(x) dx = (1/2) x^2 ln(x) - (1/2) * ∫x dx
Step 4: Evaluate the new integral:
∫x dx = (1/2) x^2
Substituting back into the equation:
∫x ln(x) dx = (1/2) x^2 ln(x) - (1/2) * (1/2) x^2 + C
Hence, the final result is:
∫x ln(x) dx = (1/2) x^2 ln(x) - (1/4) x^2 + C, where C is the constant of integration.
Note: It's always a good practice to find the final antiderivative by simplifying and rearranging the terms if possible.