Integral of sin(x)cos(x)/(sin^4(x)+cos^4(x)) (substitution)

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Integral of sin(x)cos(x)/(sin^4(x)+cos^4(x)) - How to integrate it step by step using the substitution method!

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🚶 𝐒𝐭𝐞𝐩𝐬
00:00 Multiply by 1/cos^4(x) on numerator and denominator
00:35 Rewrite expression
01:16 Write sin(x)/cos(x) as tan(x)
01:37 Substitution: u=tan(x)
01:43 Differentiate in both sides
01:50 Substitute tan(x) and (1/cos^2(x))dx
02:06 Rewrite expression
02:21 Substitution: t=u^2
02:25 Differentiate in both sides
02:35 Substitute u^2 and u*du
02:51 Rewrite expression
02:59 Integrate 1/(1+t^2)
03:07 Undo substitution: t in terms of u
03:16 Undo substitution: u in terms of x
03:27 Add integration constant +C
03:34 Final answer!

𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧 𝐦𝐞𝐭𝐡𝐨𝐝𝐬 𝐩𝐥𝐚𝐲𝐥𝐢𝐬𝐭
► Integration by parts

► Integration by substitution

► Integration by trig substitution

► Integration by Weierstrass substitution

► Integration by partial fraction decomposition

𝐅𝐨𝐥𝐥𝐨𝐰 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥𝐬 𝐅𝐨𝐫𝐘𝐨𝐮

𝐃𝐨𝐧𝐚𝐭𝐞

#integralsforyou #integrals #integrationbysubstitution
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👋 Follow @integralsforyou for a daily integral 😉

𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧 𝐦𝐞𝐭𝐡𝐨𝐝𝐬 𝐩𝐥𝐚𝐲𝐥𝐢𝐬𝐭
► Integration by parts

► Integration by substitution

► Integration by trig substitution

► Integration by Weierstrass substitution

► Integration by partial fraction decomposition


𝐅𝐨𝐥𝐥𝐨𝐰 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥𝐬 𝐅𝐨𝐫𝐘𝐨𝐮
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IntegralsForYou
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Here you have another way to get a solution:


Integral of dx =


sin(2x) = 2sin(x)cos(x) ==> (1/2)sin(2x) = sin(x)cos(x)
1^2 = (sin^2(x)+cos^2(x))^2 = sin^4(x) + 2sin^2(x)cos^2(x) + cos^4(x) = sin^4(x) + cos^4(x) + (1/2)sin^2(2x) ==> sin^4(x) + cos^4(x) = 1 - (1/2)sin^2(2x)


= Integral of (1/2)sin(2x)/( 1 - (1/2)sin^2(2x) ) dx =
= Integral of sin(2x)/(2 - sin^2(2x)) dx =
= Integral of sin(2x)/( 2 - (1-cos^2(2x)) ) dx =
= Integral of sin(2x)/(2 - 1 + cos^2(2x) ) dx =
= Integral of 1/(1+cos^2(2x)) sin(2x)dx =


Substitution:
u = cos(2x)
du = -2sin(2x) dx ==> du/-2 = sin(2x) dx


= Integral of 1/(1+u^2) du/-2 =
= (-1/2)Integral of 1/(1+u^2) du =
= (-1/2)arctan(u) =
= (-1/2)arctan(cos(2x)) + C

IntegralsForYou
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I haven’t done calculus in 3 years, I watch these simply because it’s satisfying to see. Makes you feel ever so slightly smarter c:

tovarischpootis
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You solve those things as if you already transcended above the realm of common mortals.
Amazing.

maciejstawiarz
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Hey could you please help me out with the following: integral of (4x-3) / ( sqrt(4x-x^2-3)) dx? Thank you

karolinase
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do you have any videos on how to solve by using Euler's Substitution? thank you :)

chadjohnson
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Please do: integral of 2 - sin(x)/ 2 + sin(x) dx

Angel_
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Lo de multiplicar por 1/cos(x) no tiene ninguna justificacion???
Me refiero a que como se te puede ocurrir? La unica manera es saber de memoria que esa integral se resuelve asi o hay alguna manera de darse cuenta de eso?

martinr
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Challenge to Integrals ForYou:

Integral of sqrt(tan(x))

luisg