$\int \cos ^{2}(x+1) d x$ أوجد قِيمة التكامل

Step 1 ：\(u = x + 1\)

Step 2 ：\(du = dx\)

Step 3 ：\(\int \cos^2(u) du\)

Step 4 ：Using the double angle formula: \(\cos^2(u) = \frac{1 + \cos(2u)}{2}\)

Step 5 ：\(\int \frac{1 + \cos(2u)}{2} du\)

Step 6 ：\(\frac{1}{2} \int (1 + \cos(2u)) du\)

Step 7 ：\(\frac{1}{2} (\int 1 du + \int \cos(2u) du)\)

Step 8 ：\(\frac{1}{2} (u + \frac{1}{2} \sin(2u)) + C\)

Step 9 ：Substitute back: \(\frac{1}{2} (x + 1 + \frac{1}{2} \sin(2(x + 1))) + C\)

Step 10 ：\(\boxed{\frac{1}{2} (x + 1 + \frac{1}{2} \sin(2(x + 1))) + C}\)