∫cos⁡2x dx\int \cos^{2}x \, dx
1
Use Pythagorean Identities: cos⁡2x=12+cos⁡2x2\cos^{2}x=\frac{1}{2}+\frac{\cos{2x}}{2}.
∫12+cos⁡2x2 dx\int \frac{1}{2}+\frac{\cos{2x}}{2} \, dx

2
Use Sum Rule: ∫f(x)+g(x) dx=∫f(x) dx+∫g(x) dx\int f(x)+g(x) \, dx=\int f(x) \, dx+\int g(x) \, dx.
∫12 dx+∫cos⁡2x2 dx\int \frac{1}{2} \, dx+\int \frac{\cos{2x}}{2} \, dx

3
Use this rule: ∫a dx=ax+C\int a \, dx=ax+C.
x2+∫cos⁡2x2 dx\frac{x}{2}+\int \frac{\cos{2x}}{2} \, dx

4
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
x2+12∫cos⁡2x dx\frac{x}{2}+\frac{1}{2}\int \cos{2x} \, dx

5
Use Integration by Substitution on ∫cos⁡2x dx\int \cos{2x} \, dx.
Let u=2xu=2x, du=2 dxdu=2 \, dx, then dx=12 dudx=\frac{1}{2} \, du

6
Using uu and dudu above, rewrite ∫cos⁡2x dx\int \cos{2x} \, dx.
∫cos⁡u2 du\int \frac{\cos{u}}{2} \, du

7
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
12∫cos⁡u du\frac{1}{2}\int \cos{u} \, du

8
Use Trigonometric Integration: the integral of cos⁡u\cos{u} is sin⁡u\sin{u}.
sin⁡u2\frac{\sin{u}}{2}

9
Substitute u=2xu=2x back into the original integral.
sin⁡2x2\frac{\sin{2x}}{2}

10
Rewrite the integral with the completed substitution.
x2+sin⁡2x4\frac{x}{2}+\frac{\sin{2x}}{4}

11
Add constant.
x2+sin⁡2x4+C\frac{x}{2}+\frac{\sin{2x}}{4}+C

Done
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