Question

Q. Find the reduction formula of ʃcos12xdx

Answer 1

Answer on Question #53131 – Math – Integral Calculus

Q. Find the reduction formula of $\int \cos^1 2(x) \, dx$.

Solution

$$\begin{aligned} I_n = & \int \cos^n x \, dx = \int \cos^{n-1} x \cos x \, dx = \int \cos^{n-1} x \, d(\sin x) = \\ &= \cos^{n-1} x \sin x + (n - 1) \int \cos^{n-2} x \sin^2 x \, dx = \\ &= \cos^{n-1} x \sin x + (n - 1) \int \cos^{n-2} x (1 - \cos^2) x \, dx = \\ &= \cos^{n-1} x \sin x + (n - 1) I_{n-2} + (n - 1) I_n \rightarrow \\ &\rightarrow I_n = \frac{1}{n} \cos^{n-1} x \sin x + \frac{n - 1}{n} I_{n-2} \end{aligned}$$

Thus,

$$\begin{aligned} I_{12} = &\frac{1}{12} \cos^{11} x \sin x + \frac{11}{12} I_{10} = \frac{1}{12} \cos^{11} x \sin x + \frac{11}{12} \left(\frac{1}{10} \cos^9 x \sin x + \frac{9}{10} I_8\right) = \\ &= \frac{1}{12} \cos^{11} x \sin x + \frac{11}{120} \cos^9 x \sin x + \frac{99}{120} \left(\frac{1}{8} \cos^7 x \sin s + \frac{7}{8} I_6\right) = \\ &= \frac{1}{12} \cos^{11} x \sin x + \frac{11}{120} \cos^9 x \sin x + \frac{99}{960} \cos^7 x \sin x + \frac{693}{960} \left(\frac{1}{6} \cos^5 x \sin x + \frac{5}{6} I_4\right) = \\ &= \frac{1}{12} \cos^{11} x \sin x + \frac{11}{120} \cos^9 x \sin x + \frac{33}{320} \cos^7 x \sin x + \frac{231}{1920} \cos^5 x \sin x + \\ &\quad + \frac{1155}{1920} \left(\frac{1}{4} \cos^3 x \sin x + \frac{3}{4} I_2\right) = \\ &= \frac{1}{12} \cos^{11} x \sin x + \frac{11}{120} \cos^9 x \sin x + \frac{33}{320} \cos^7 x \sin x + \frac{231}{1920} \cos^5 x \sin x + \\ &\quad + \frac{1155}{7680} \cos^3 x \sin x + \frac{3465}{7680} \left(\frac{1}{2} \cos x \sin x + \frac{1}{2} I_0\right) \end{aligned}$$

But $I_0 = \int dx = x + c$

So

$$\begin{array}{l} \int \cos^{12} x \, dx = \\ = \frac{\sin 2x}{30720} \left(1280 \cos^{10} x + 1408 \cos^{8} x + 1584 \cos^{6} + 1848 \cos^{4} x + 2310 \cos^{2} x + 3465\right) + \\ + \frac{3465}{15360} x + c, \end{array}$$

where $c$ is an arbitrary real constant.

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