Using:

sin⁴x + cos⁴x = (sin²x + cos²x)² − 2sin²x cos²x

= 1 − 2sin²x cos²x

= 1 − 1/2 sin²2x

Average value of sin²2x over a complete period is:

1/2

Therefore average value of integrand:

1 − 1/2 × 1/2 = 3/4

Length of interval:

20π

Integral:

= (3/4)(20π)

= 15π

Final Answer

Option (1) 15π