The value of $\int_{0}^{20 \pi}\left(\sin ^{4} \mathrm{x}+\cos ^{4} \mathrm{x}\right) \mathrm{dx}$ is equal to:

JEE Main · 2026 · 02 Apr 2026 · Shift 2 · Definite Integrals

The value of

\int_{0}^{20 π} (sin^{4} x + cos^{4} x) dx

is equal to:

Answer: (C)

Using

sin^{4} x + cos^{4} x = 1 - 2 sin^{2} x cos^{2} x = 1 - \frac{1}{2} sin^{2} 2 x

I = \int_{0}^{20 π} (1 - \frac{1}{2} sin^{2} 2 x) d x = 20 π - \frac{1}{2} \int_{0}^{20 π} sin^{2} 2 x d x .

Over the full range the mean value of

sin^{2} 2 x

\frac{1}{2}

, so

\int_{0}^{20 π} sin^{2} 2 x d x = \frac{1}{2} (20 π) = 10 π

. Therefore

I = 20 π - 5 π = 15 π

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