The following integrals require a preliminary step such as a change of variables

iohanetc 2021-10-17 Answered
The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.
cosθ(sin3θ4sinθ)dθ
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Expert Answer

Brittany Patton
Answered 2021-10-18 Author has 100 answers
We first use substitution
Let,
z=sinθ
dz=cosθdθ
cosθ(sin3θ4sinθ)dθ
=1(z34z)dz
=1z(z24)dz=1z(z+2)(z2)dz
Using partial decomposition we get
1z(z+2)(z2)=Az+Bz+2+Cz2
1z(z+2)(z2)=A(z+2)(z2)+Bz(z2)+Cz(z+2)z(z+2)(z2)
1=A(z+2)(z2)+Bz(z2)+Cz(z+2)
1=Az24A+Bz2+Cz2+2Cz
1=(A+B+C)z2+(2B+2C)z4A
A+B+C=0
A+B+C=0
2B+2C=0
4A=1
A=14,B=18,C=18
1z(z+2)(z2)dz
=(14z+18z+2+18z2)dz

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