# Integral Of Trig Functions

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Some integration is the inverse differentiation; the integrals of some elementary functions follow from the standard results of differential calculus. Following are the few examples of integral trigonometric functions.

Question: - Evaluate
1. ∫ cos ^2 2 x dx
2. ∫ sin ^3 x dx
3. ∫ sin 3 x sin 4 x d x
Solution: -∫ cos ^2 2 x dx = ½  ∫ 2 cos ^2 2x dx

= ½ ∫ (1 + cos 4 x)

= ½ [∫ dx + ∫ cos 4x]

= ½ [ x + (sin 4x)/4] + c                     Where c is a constant.

2.Since, sin 3 x = 3 sin x – 4 sin ^3 x,

Therefore, 4 sin ^3 x = 3 sin x – sin 3 x

Hence, ∫ sin ^3 x dx = ¼ ∫ 4 sin ^3 x dx

= ¼ ∫ (3 sin x – sin 3 x) dx

= ¼ [3 ∫ sin x dx - ∫ sin 3 x dx]

= ¼ [3 (- cos x) – (- cos 3 x / 3) ] + k

= -3/4 cos x + 1 / 12 cos 3 x + k

3.∫ sin 3 x sin 4 x d x = ½ ∫ 2 sin 3 x sin 4 x d x

= ½ ∫ (cos x – cos 7 x) d x

= ½ [∫ cos x dx - ∫ cos 7 x dx]

= ½ [sin x – sin 7x / 7] + k