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Question : 77 of 160
Marks:
+1 ,
-0
Solution:
‌ ‌ Let ‌ I = x 4 ( 4 − x 2 ) 7 ∕ 2 ‌ dx ‌ I = 2 ‌ x 4 ( 4 − x 2 ) 7 ∕ 2 ‌ dx ‌ ‌ ‌ [ ∵ f ( x ) ‌ is even function ‌ ] ‌ ‌ put ‌ x = 2 s i n ‌ θ ⇒ dx = 2 ‌ cos ‌ θ ‌ d ‌ θ ‌ ‌ when ‌ x = 0 , θ = 0 ‌ and ‌ x = 2 , θ = ‌ ‌ ∴ I = 2 ‌ 2 4 s i n ‌ 4 θ ( 2 ‌ cos ‌ θ ) 7 ⋅ 2 ‌ cos ‌ θ ‌ d ‌ θ ‌ I = 2 ⋅ 2 4 ⋅ 2 8 ‌ s i n ‌ 4 θ cos 8 θ d θ ‌ I = 2 13 . ‌ ‌ ( 4 − 1 ) ( 4 − 3 ) ( 8 − 1 ) ( 8 − 3 ) ( 8 − 5 ) ( 8 − 7 ) 12 × 10 × 8 × 6 × 4 × 2
× ‌ (by Wall's formulae)
‌ = ‌ 2 13 × 3 × 1 × 7 × 5 × 3 × 1 × 12 × 10 × 8 × 6 × 4 × 2
‌ = ‌ 2 6 × 3 × 7 × 5 × 3 × π 6 × 5 × 4 × 3 × 2
= 28 π
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