Integral of 1/(1+x^2)^2 (substitution)

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285.7 هزار بار بازدید - 8 سال پیش - ✍🏼
✍🏼 https://integralsforyou.com - Integral of 1/(1+x^2)^2 - How to integrate it step by step using integration by substitution!

✅ 𝐃𝐞𝐫𝐢𝐯𝐚𝐭𝐢𝐯𝐞 𝐭𝐨 𝐜𝐡𝐞𝐜𝐤 𝐭𝐡𝐞 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧
Derivative of (1/2)(arctan(x) + x/(1+x^2)) = Derivative of (1/2)(arctan(x) + x/(1+...

🚶 𝐒𝐭𝐞𝐩𝐬
00:00 Substitution: x=tan(u)
00:12 Differentiate in both sides
00:29 Substitute x and dx
00:51 Simplify expression 1 + sin^2(u)/cos^2(u)
02:05 Simplify expression 1/cos^2(u)
02:20 Simplify expression cos^4(u)/cos^2(u)
02:36 Apply cos^2(u) = (1/2)(1+cos(2u))
02:55 Split into two integrals
03:12 Multiply and divide by 2
03:21 Integrate 1 and 2cos(2u)
03:43 Apply sin(2u) = 2sin(u)cos(u)
04:27 Undo substitution: u in terms of x
05:05 Apply sin(arctan(x)) and cos(arctan(x)) formulas
05:45 Add integration constant +C
05:54 Final answer!


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8 سال پیش در تاریخ 1395/01/11 منتشر شده است.
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