\[\int \cos^{4}x \, dx\]
1
Use Trigonometric Reduction Formulas.
cos⁡3xsin⁡x4+34∫cos⁡2x dx\frac{\cos^{3}x\sin{x}}{4}+\frac{3}{4}\int \cos^{2}x \, dx

2
Use Pythagorean Identities: cos⁡2x=12+cos⁡2x2\cos^{2}x=\frac{1}{2}+\frac{\cos{2x}}{2}.
cos⁡3xsin⁡x4+34∫12+cos⁡2x2 dx\frac{\cos^{3}x\sin{x}}{4}+\frac{3}{4}\int \frac{1}{2}+\frac{\cos{2x}}{2} \, dx

3
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.
cos⁡3xsin⁡x4+34(∫12 dx+∫cos⁡2x2 dx)\frac{\cos^{3}x\sin{x}}{4}+\frac{3}{4}(\int \frac{1}{2} \, dx+\int \frac{\cos{2x}}{2} \, dx)

4
Use this rule: ∫a dx=ax+C\int a \, dx=ax+C.
cos⁡3xsin⁡x4+34(x2+∫cos⁡2x2 dx)\frac{\cos^{3}x\sin{x}}{4}+\frac{3}{4}(\frac{x}{2}+\int \frac{\cos{2x}}{2} \, dx)

5
Use Constant Factor Rule: ∫cf(x) dx=c∫f(x) dx\int cf(x) \, dx=c\int f(x) \, dx.
cos⁡3xsin⁡x4+34(x2+12∫cos⁡2x dx)\frac{\cos^{3}x\sin{x}}{4}+\frac{3}{4}(\frac{x}{2}+\frac{1}{2}\int \cos{2x} \, dx)

6
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

7
Using uu and dudu above, rewrite ∫cos⁡2x dx\int \cos{2x} \, dx.
∫cos⁡u2 du\int \frac{\cos{u}}{2} \, du

8
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

9
Use Trigonometric Integration: the integral of cos⁡u\cos{u} is sin⁡u\sin{u}.
sin⁡u2\frac{\sin{u}}{2}

10
Substitute u=2xu=2x back into the original integral.
sin⁡2x2\frac{\sin{2x}}{2}

11
Rewrite the integral with the completed substitution.
cos⁡3xsin⁡x4+34(x2+sin⁡2x4)\frac{\cos^{3}x\sin{x}}{4}+\frac{3}{4}(\frac{x}{2}+\frac{\sin{2x}}{4})

12
Add constant.
cos⁡3xsin⁡x4+34(x2+sin⁡2x4)+C\frac{\cos^{3}x\sin{x}}{4}+\frac{3}{4}(\frac{x}{2}+\frac{\sin{2x}}{4})+C

Done
Copy Link
How can we make this solution more helpful?
Cymath foriOSAndroid