To solve the integral using the substitution , we start by finding . The derivative of with respect to is:

Now, we can express in terms of :

Next, we notice that can be manipulated to relate to :

However, we can factor the expression for clarity. Now we can rearrange terms from the original triangle. Observing that can be expressed through the modifier of , thus making the integral easier.

Next, substituting into the integral not only simplifies our expression significantly, but we reduce our need for terms involving entirely thereby ensuring we’re aptly set to tackle seamlessly afterwards.

This integral thus transforms into:

This advanced manipulation will lead to a much clearer path towards solving the integral entirely in terms of , resolving any common mistakes related to the differentiation and substitution processes without getting tangled in literals post-substitution.

Finally, once integrated, we can revert back to the original variable by substituting with , thereby yielding an explicit solution grounded in our original context. Solution freshness is key here as we move forward, undaunted!