Question

∫ cos |x| dx=? . limit is from - 2π to π

Answer 1

Question #51707, Math, Integral Calculus

$$\int_{-2\pi}^{\pi} \cos|x| \, dx$$

Solution

$$\begin{array}{l} |x| = \left\{ \begin{array}{l} - x, x < 0 \\ x, x \geq 0 ; \end{array} \right. ; \\ \cos(-x) = \cos x. \end{array}$$

Therefore

$$\begin{array}{l} \int_{-2\pi}^{\pi} \cos|x| \, dx = \int_{-2\pi}^{0} \cos|x| \, dx + \int_{0}^{\pi} \cos|x| \, dx = \int_{-2\pi}^{0} \cos(-x) \, dx + \int_{0}^{\pi} \cos x \, dx = \int_{-2\pi}^{0} \cos x \, dx + \int_{0}^{\pi} \cos x \, dx = \\ = \int_{-2\pi}^{\pi} \cos x \, dx = \sin x \Big|_{-2\pi}^{\pi} = \sin \pi - \sin(-2\pi) = 0. \end{array}$$

Answer: 0.

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