The value of ∫ π 6 π 3 e sec 2 ⁡ x sin ⁡ x cos 3 ⁡ x d x is equal to

Question

The value of ∫ π 6 π 3 e sec 2 ⁡ x sin ⁡ x cos 3 ⁡ x d x is equal to, 1 2 e 4, 1 2 e 4 3, 1 2 e 4 - e 4 3, 1 2 e 2 - 1

MARKS App · Accepted Answer

Let, I = ∫ e s e c 2 x sin ⁡ x cos 3 ⁡ x d x ⇒ I = ∫ e sec 2 ⁡ x tanx · sec 2 ⁡ x d x ⇒ I = ∫ e · e tan 2 ⁡ x tan ⁡ x s e c 2 d x Substituting, tan 2 ⁡ x = t 2 tan ⁡ x sec 2 ⁡ x d x = d t ⇒ I = 1 2 ∫ e · e t d t I = e 2 e t + C I = e 2 e tan 2 ⁡ x + C = e sec 2 ⁡ x 2 + C Ι = 1 2 e sec 2 ⁡ x π 6 π 3 = 1 2 e 4 - e 4 3

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