$\int_{0}^{20\pi}(|\sin x|+|\cos x|)^{2} dx$ is equal to:

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Definite Integrals

\int_{0}^{20 π} (∣ sin x ∣ + ∣ cos x ∣)^{2} d x

is equal to:

Answer: (D)

Using the property that the integrand has period

\frac{π}{2}

I = \int_{0}^{20 π} (∣ sin x ∣ + ∣ cos x ∣)^{2} d x = 40 \int_{0}^{π /2} (sin x + cos x)^{2} d x

= 40 \int_{0}^{π /2} (1 + sin 2 x) d x

= 40 [x - \frac{c o s 2 x}{2}]_{0}^{π /2}

= 40 [\frac{π}{2} - \frac{- 1}{2} - 0 + \frac{1}{2}]

= 40 [\frac{π}{2} + \frac{1}{2}] = 20 (π + 2)

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Single correct option

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