Example 27 (iv) - Evaluate sin3 2t cos 2 t dt - Chapter 7 Class 12 CBSE NCERT Math - Definate Integration - By Substitution

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  1. Chapter 7 Class 12 Integrals
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=﷐𝐥𝐨𝐠﷮﷐﷐𝟑𝟐﷮𝟐𝟕﷯﷯﷯ Example 27 Evaluate the following integrals: (iv) ﷐0﷮﷐𝜋﷮4﷯﷮﷐﷐sin﷮3﷯﷮2𝑡﷯﷐cos﷮2﷯𝑡﷯ 𝑑𝑡 ﷐0﷮﷐𝜋﷮4﷯﷮𝑠𝑖﷐𝑛﷮3﷯2𝑡 𝑐𝑜𝑠 2𝑡 𝑑𝑡﷯ Step 1 :- F﷐𝑥﷯=﷐﷮﷮𝑠𝑖﷐𝑛﷮3﷯2𝑡 𝑐𝑜𝑠 2𝑡 𝑑𝑡﷯ Let s𝑖𝑛 2𝑡=𝑢 Differentiating w.r.t.𝑥 ﷐𝑑(﷐sin﷮2𝑡﷯)﷮𝑑𝑡﷯=﷐𝑑𝑢﷮𝑑𝑡﷯ 2c𝑜𝑠 2𝑡 =﷐𝑑𝑢﷮𝑑𝑡﷯ 𝑑𝑡=﷐𝑑𝑢﷮2 𝑐𝑜𝑠 2𝑡﷯ Hence the integrate ﷐﷮﷮𝑠𝑖﷐𝑛﷮3﷯ 2𝑡 𝑐𝑜𝑠 2𝑡 𝑑𝑡﷯=﷐﷮﷮﷐𝑢﷮3﷯ 𝑐𝑜𝑠 2𝑡 × ﷐𝑑𝑢﷮2 𝑐𝑜𝑠 2𝑡﷯﷯ =﷐1﷮2﷯﷐﷮﷮﷐𝑢﷮3﷯ 𝑑𝑢﷯ =﷐1﷮2﷯ ﷐﷐𝑢﷮3+1﷯﷮3+1﷯=﷐1﷮2﷯ ﷐﷐𝑢﷮4﷯﷮4﷯= ﷐﷐𝑢﷮4﷯﷮8﷯ Putting back 𝑢=𝑠𝑖𝑛 2𝑡 =﷐1﷮8﷯𝑠𝑖﷐𝑛﷮4﷯ 2𝑡 Hence F﷐𝑡﷯=﷐1﷮8﷯𝑠𝑖﷐𝑛﷮4﷯ 2𝑡 Step 2 :- ﷐0﷮﷐𝜋﷮4﷯﷮𝑠𝑖﷐𝑛﷮3﷯ 2𝑡 𝑐𝑜𝑠 2𝑡=𝐹﷐﷐𝜋﷮4﷯﷯−𝐹﷐0﷯﷯ =﷐1﷮8﷯𝑠𝑖﷐𝑛﷮4﷯ 2﷐﷐𝜋﷮4﷯﷯−﷐1﷮8﷯𝑠𝑖﷐𝑛﷮4﷯ 2﷐0﷯ =﷐1﷮8﷯𝑠𝑖﷐𝑛﷮4﷯﷐𝜋﷮2﷯−﷐1﷮8﷯𝑠𝑖﷐𝑛﷮4﷯ ﷐0﷯ =﷐1﷮8﷯ ×﷐1﷮4﷯−﷐1﷮8﷯ ×﷐0﷮4﷯ =﷐1﷮8﷯ ×1−0 =﷐1﷮8﷯

Chapter 7 Class 12 Integrals
Concept wise

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