Evaluate the following integral. ∫x3sin⁡(x4)cos5(x4)dx

Question

Evaluate the following integral.  ∫x3sin⁡(x4)cos5(x4)dx

Joseph Fair · Accepted Answer

Step 1  Consider the given integral  ∫x3sin⁡(x4)cos5(x4)dx  Step 2  Use substitution method to evaluate the integral  Let u=cos⁡(x4)  Then du=−4x3sin⁡(x4)dx  ⇒x3sin⁡(x4)dx=−14du  Step 3  Then the integral becomes  ∫x3sin⁡(x4)cos5(x4)dx=∫u5(−14)du  =−14∫u5du  =−14[u66]  =−u624  Substituting u=cos⁡(x4), we get −cos6(x4)24  ∴∫x3sin⁡(x4)cos5(x4)dx=−cos6(x4)24+C,  where C is the constant of integration.

Heather Fulton · Accepted Answer

∫x3sin⁡(x4)cos5(x4)dx  4⋅x3dx=d(x4),t=x4  ∫sin⁡(t)⋅cos⁡(t)54dt  (−sin⁡(x))dx=d(cos⁡(x)),u=cos⁡(x)  ∫(−u54)du  ∫−u54du=−u624+C  −cos⁡(t)624+C  −cos⁡(x4)624+C

Vasquez · Accepted Answer

∫x3∗sin⁡(x4)cos5⁡(x4)dxTransform∫−t54dt−14∗∫t5dt−14∗t66Substitute back−14∗cos⁡(x4)66Multiply−cos⁡(x4)624Add CSolution−cos⁡(x4)624+C

Evaluate the following integral. ∫x3sin⁡(x4)cos5(x4)dx

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