Question

Find the integrals of the function: $cos^4\ 2x$

Answer 1

wait for awhile ..here u are ..miss

Answer 2

we have to find ∫cos⁴(2x) dx

first of all resolve the cos⁴(2x) into simpler form.

we know, cos²Φ = (1 + cos2Φ)/2

so, cos⁴(2x) = [cos²(2x) ]²

= [{1 + cos2(2x)}/2 ]²

= [(1 + cos4x)/2 ]²

= (1 + cos4x)²/4

= {1 + cos²(4x) + 2cos(4x)}/4

= 1/4 + cos²(4x)/4 + cos(4x)/2

= 1/4 + {1 + cos2(4x)}/8 + cos(4x)/2

= 1/4 + 1/8 + cos(8x)/8 + cos(4x)/2

= 3/8 + cos(8x)/8 + cos(4x)/2

now, ∫cos⁴(2x) dx = ∫[3/8 + cos(8x)/8 + cos(4x)/2 ] dx

= 3/8∫dx + 1/8∫cos(8x) dx + 1/2∫ cos(4x) dx

= 3x/8 + 1/8 sin(8x)/8 + 1/2 sin(4x)/4 + K, where K is constant

= 3x/8 + 1/64 sin(8x) + 1/8 sin(4x) + K,