Evaluate the indefinite integral. \int (\cot x)\ln (\sin x)dx

Pretoto4o 2021-11-19 Answered
Evaluate the indefinite integral.
(cotx)ln(sinx)dx
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Expert Answer

Lounctirough
Answered 2021-11-20 Author has 14 answers
Step 1
we have to evaluate the integral
(cotx)ln(sinx)dx...(1)
let cotx=zlog(sinx)dx=dz
Step 2
substituting it in the equation (1) we get
zdz
integrating it we get
[z22]
putting the value of z we get
[cot2x2]
therefore the answer of (cotx)ln(sinx)dx is cot2x2
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Marlene Broomfield
Answered 2021-11-21 Author has 15 answers
Step 1: Remove parentheses.
cotxln(sinx)dx
Step 2: Use Integration by Parts on cotxln(sinx)dx
Let u=ln(sinx),dv=cotx,du=cosxsinxdx,v=ln(sinx)
Step 3: Substitute the above into uvvdu.
ln(sinx)2ln(sinx)cosxsinxdx
Step 4: Use Integration by Substitution on ln(sinx)cosxsinxdx.
lnuudu
Step 6: Use Integration by Substitution.
Let w=lnu,dw=1udu
Step 7: Using w and dw above, rewrite lnuudu.
wdw
Step 8: Use Power Rule: xndx=xn+1n+1+C.
w22
Step 9: Substitute w=lnu back into the original integral.
lnu22
Step 10: Substitute u=sinx back into the original integral.
ln(sinx)22
Step 11: Rewrite the integral with the completed substitution.
ln(sinx)22
Step 12: Add constant.
ln(sinx)22+C
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