Question

Find the reduction formula of ʃcos8xdx.

Answer 1

Answer on Question #53129 – Math – Integral Calculus

Find the reduction formula of $\int \cos^8 x \, dx$.

Solution

$$\int \cos^8 x \, dx = \int \frac{1 + \cos^4 2x}{2} \, dx = \int \left(\frac{1}{2} + \frac{1 + \cos^2 4x}{4}\right) dx = \int \left(\frac{1}{2} + \frac{1}{4} + \frac{1 + \cos 8x}{8}\right) dx = \int \left(\frac{7}{8} + \frac{1}{8} \cos 8x\right) dx = \frac{7}{8}x + \frac{1}{64} \sin 8x + C = \frac{56 + \sin 8x}{64} + C,$$

where $C$ is an arbitrary real constant.

Answer: $\frac{56 + \sin 8x}{64} + C$.

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