Sure! Let's solve the expression step by step:
Given:
[tex]\[ d + \int d \, \mathrm{d}d + d + d \][/tex]
Step 1: Evaluate the integral [tex]\(\int d \, \mathrm{d}d\)[/tex].
The integral of [tex]\(d\)[/tex] with respect to [tex]\(d\)[/tex] is:
[tex]\[ \int d \, \mathrm{d}d = \frac{d^2}{2} \][/tex]
Step 2: Substitute the integral back into the original expression. Now, our expression looks like:
[tex]\[ d + \frac{d^2}{2} + d + d \][/tex]
Step 3: Combine like terms. We have three terms involving [tex]\(d\)[/tex] and one term involving [tex]\(\frac{d^2}{2}\)[/tex]:
[tex]\[ \frac{d^2}{2} + d + d + d \][/tex]
Step 4: Simplify the terms involving [tex]\(d\)[/tex]. Adding [tex]\(d\)[/tex] three times gives us [tex]\(3d\)[/tex]:
[tex]\[ \frac{d^2}{2} + 3d \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ \boxed{\frac{d^2}{2} + 3d} \][/tex]
