Use a change of variables to evaluate the following integral. ∫−(cos7⁡x−5cos5⁡x−cos⁡x)sin⁡xdx

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Use a change of variables to evaluate the following integral.  ∫−(cos7⁡x−5cos5⁡x−cos⁡x)sin⁡xdx

Faiza Fuller · Accepted Answer

Step 1 The given integral is, ∫−(cos7⁡x−5cos5⁡x−cos⁡x)sin⁡xdx Step 2 The above integral can be split into the following integrals. I=∫−cos7⁡xsin⁡xdx+∫5cos5⁡xsin⁡xdx+∫cos⁡xsin⁡xdx Step 3 The first integral is evaluated using the substitution as follows. ∫−cos7⁡xsin⁡xdx=−∫−u7du(Use the substitution,u=cos⁡x) =u88+c1 =cos8⁡x8+c1 Step 4 The second integral is evaluated using the substitution as follows. ∫5cos5⁡xsin⁡xdx=5∫cos5⁡xsin⁡xdx =−5∫u5du (Use the substitution, u=cos⁡x) =−56u6+c2 =−56cos6⁡x+c2 Step 5 The third integral is evaluated using the substitution as follows. ∫cos⁡xsin⁡xdx=∫udu (Use the substitution, u=sin⁡x) =u22+c3 =sin2⁡x2+c3 Step 6 Thus, the integral ∫ can be written as follows. I=cos8⁡x8+(−56)cos6⁡x+sin2⁡x2+C

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

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