Integral of sqrt(tan(x)) (substitution)

🍏 𝐀𝐧 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥 𝐚 𝐝𝐚𝐲 𝐤𝐞𝐞𝐩𝐬 𝐭𝐡𝐞 𝐝𝐨𝐜𝐭𝐨𝐫 𝐚𝐰𝐚𝐲
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✅ 𝐃𝐞𝐫𝐢𝐯𝐚𝐭𝐢𝐯𝐞 𝐭𝐨 𝐜𝐡𝐞𝐜𝐤 𝐭𝐡𝐞 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧
Derivative of (1/√2)arctan((tanx-1)/(√2 √tanx)) + ln((tanx+1)/√tanx - √2) - ln((tanx+1)/√tanx + √2) =    • Derivative of (1/√2)arctan((tanx-1)/(...  

💪 𝐇𝐨𝐰 𝐭𝐨 𝐜𝐨𝐥𝐥𝐚𝐛𝐨𝐫𝐚𝐭𝐞
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🎓 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧 𝐦𝐞𝐭𝐡𝐨𝐝𝐬 𝐩𝐥𝐚𝐲𝐥𝐢𝐬𝐭𝐬
   • 🧑‍🔧 Integration by parts  
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🚶 𝐒𝐭𝐞𝐩𝐬
𝟏. 𝐒𝐮𝐛𝐬𝐭𝐢𝐭𝐮𝐭𝐢𝐨𝐧 𝐭 = √𝐭𝐚𝐧(𝐱)
00:00 Substitution: t = √tan(x)
00:10 Differentiate in both sides
00:44 tan(x) in terms of t
00:51 tan^2(x) in terms of t
01:10 dx in terms of t
01:20 Substitute √tan(x) and dx
01:35 Multiply t by 2t
01:44 Write 2t^2 as t^2 + 1 + t^2 - 1
02:01 Split into two integrals
02:21 Multiply by 1/t^2 on numerator and denominator

𝟐. 𝐒𝐮𝐛𝐬𝐭𝐢𝐭𝐮𝐭𝐢𝐨𝐧 𝐯 = 𝐭 + 𝟏/𝐭
03:03 Substitution: v = t + 1/t
03:12 Differentiate in both sides
03:22 v^2 in terms of t
03:33 t^2 + 1/t^2 in terms of v
03:44 Substitute t^2 + 1/t^2 and (1 + 1/t^2)dt

𝟑. 𝐒𝐮𝐛𝐬𝐭𝐢𝐭𝐮𝐭𝐢𝐨𝐧 𝐮 = 𝐭 - 𝟏/𝐭
04:04 Substitution: u = t - 1/t
04:11 Differentiate in both sides
04:19 u^2 in terms of t
04:29 t^2 + 1/t^2 in terms of u
04:41 Substitute t^2 + 1/t^2 and (1 - 1/t^2)dt

𝟒. 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥 𝐨𝐟 𝟏/(𝟐+𝐯^𝟐)
05:06 Prepare for the integral of 1/(2+v^2)
05:29 Take out the 2 as common factor
05:41 Take the 1/2 outside the integral and write v^2/2 as (v/√2)^2
05:56 Substitution: r = v/√2
06:03 Differentiate in both sides
06:09 dv in terms of r
06:16 Substitute v/√2 and dv
06:37 Take the √2 outside the integral
06:47 Simplify √2/2 and integrate 1/(1+r^2)
06:56 Undo substitution: v in terms of r
07:34 Answer for the integral of 1/(2+v^2)

𝟓. 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥 𝐨𝐟 𝟏/(𝐮^𝟐-𝟐)
07:43 Prepare for the integral of 1/(u^2-2)
08:22 Partial fraction decomposition for 1/(u^2-2)
08:26 Write u^2-2 as (u-√2)(u+√2)
08:40 Write 1/(u-√2)(u+√2) = A/(u-√2) + B/(u+√2)
08:53 Write A/(u-√2) + B/(u+√2) as [(A+B)u + √2(A-B)]/(u^2-2)
10:06 Move paper to have more space :)
10:20 Deduce 0 = A+B and 1 = √2(A-B) from 0u+1 = (A+B)u + √2(A-B)
10:34 Look for A and B values
10:55 B = -1/2√2
11:07 A = 1/2√2
11:20 Write 1/(u^2-2) as (1/2√2)/(u-√2) - (1/2√2)/(u+√2)
12:02 Split the integral of 1/(u^2-2) into two integrals
12:23 Integrate 1/(u-√2) and 1/(u+√2)
12:42 Answer for the integral of 1/(2+v^2)

𝟔. 𝐔𝐧𝐝𝐨 𝐬𝐮𝐛𝐬𝐭𝐢𝐭𝐮𝐭𝐢𝐨𝐧𝐬 𝐚𝐧𝐝 𝐟𝐢𝐧𝐚𝐥 𝐚𝐧𝐬𝐰𝐞𝐫
13:23 Undo substitution: v in terms of t
13:38 Undo substitution: u in terms of t
14:11 Undo substitution: t in terms of x
15:23 Add integration constant +C
15:42 Final answer!
15:48 See more!

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