Use a change of variables to evaluate the following integral. \int-(\cos^{7}x-5\cos^{5}x-\cos x)\sin x dx

a2linetagadaW

a2linetagadaW

Answered question

2021-05-23

Use a change of variables to evaluate the following integral.
(cos7x5cos5xcosx)sinxdx

Answer & Explanation

Faiza Fuller

Faiza Fuller

Skilled2021-05-24Added 108 answers

Step 1
The given integral is,
(cos7x5cos5xcosx)sinxdx
Step 2
The above integral can be split into the following integrals.
I=cos7xsinxdx+5cos5xsinxdx+cosxsinxdx
Step 3
The first integral is evaluated using the substitution as follows.
cos7xsinxdx=u7du(Use the substitution,u=cosx)
=u88+c1
=cos8x8+c1
Step 4
The second integral is evaluated using the substitution as follows.
5cos5xsinxdx=5cos5xsinxdx
=5u5du (Use the substitution, u=cosx)
=56u6+c2
=56cos6x+c2
Step 5
The third integral is evaluated using the substitution as follows.
cosxsinxdx=udu (Use the substitution, u=sinx)
=u22+c3
=sin2x2+c3
Step 6
Thus, the integral can be written as follows.
I=cos8x8+(56)cos6x+sin2x2+C

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