Question 12 - CBSE Class 12 Sample Paper for 2026 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

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Question 12 If ∫𝑥^3 sin^4 (𝑥^4 )cos(𝑥^4 )𝑑𝑥=𝑎sin^5 (𝑥^4 )+C, then 𝑎 is equal to (A) −1/10 (B) 1/20 (C) 1/4 (D) 1/5Finding ∫𝑥^3 sin^4 (𝑥^4 )cos(𝑥^4 )𝑑𝑥 Let 𝐭=𝐬𝐢𝐧⁡(𝒙^𝟒 ) Differentiating t 𝒅𝒕/𝒅𝒙=〖𝒄𝒐𝒔 〗⁡(𝒙^𝟒 ) × (𝒙^𝟒 )^′ 𝑑𝑡/𝑑𝑥=〖𝑐𝑜𝑠 〗⁡(𝑥^4 ) × 4𝑥^3 𝑑𝑡=〖𝑐𝑜𝑠 〗⁡(𝑥^4 )4𝑥^3 𝑑𝑥 𝒅𝒕/𝟒=〖𝒄𝒐𝒔 〗⁡(𝒙^𝟒 ) 𝒙^𝟑 𝒅𝒙 Now, ∫𝑥^3 sin^4 (𝑥^4 )cos(𝑥^4 )𝑑𝑥=∫1▒(𝒕^𝟒 𝒅𝒕)/𝟒 =1/4 ∫1▒〖𝑡^4 𝑑𝑡〗 =1/4 ×𝑡^5/5+𝐶 =𝟏/𝟐𝟎 𝒕^𝟓+𝐶 Putting back 𝑡=𝑠𝑖𝑛⁡(𝑥^4 ) =𝟏/𝟐𝟎 〖𝒔𝒊𝒏〗^𝟓 (𝒙^𝟒 ) +𝑪 Comparing with 𝒂〖𝒔𝒊𝒏〗^𝟓 (𝒙^𝟒 )+𝐂, then 𝑎 is equal to 𝟏/𝟐𝟎 So, the correct answer is (B)