Evaluate the following Integral: ∫ x log(1 + 2x) dx, x ∈ [0, 1]

Question

Evaluate the following Integral:

\(\int\limits_0^1 \)x log (1 + 2x) dx

Answer 1

Now let us use integration by parts

For this we have to apply integration by parts

Let u and v be two functions then

∫udv = uv - ∫vdu

To choose the first function u we use “ILATE” rule

That is

I=inverse trigonometric function

L=logarithmic function

A=algebraic function

T=trigonometric functions

E=exponential function

So in this preference, the first function is chosen to make the integration simpler.

Now, In the given question x² is an algebraic function and it is chosen as u(A comes first in “ILATE” rule)

So first let us integrate the equation and then let us substitute the limits in it.