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How do you integrate #((cosx)^3 sinx )/(4sqrt{1-(cosx)^4})dx#?

Answer 1

Christopher Agresta

This integral is quite simple to understand.

#1/4int(cos^3(x)*sin(x))/(sqrt(1-cos^4(x)))dx #

Let's #t=cos(x)#

So #dt = -sin(x)dx#

#1/2intt^3/(2(sqrt(1-t^4)))d(-t) = 1/2int(-t^3)/(2sqrt(1-t^4))dt #

Let's #u = 1-t^4#

So #du = -4t^3dt#

#=>1/8int(-4t^3)/(2sqrt(1-t^4))dt#

#=>1/8int1/(2sqrt(u))du#

#= 1/8[sqrt(u)]#

Substitute back for #u = 1-t^4# and #t=cos(x)#

#= 1/8[sqrt(1-cos^4(x))] + C#

With habits you can directly do #t = 1 - cos^4(x)#

#dt = 4sin(x)cos^3(x)#

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Answer 2

Luke Alvarez

To integrate ( \frac{{(\cos x)^3 \sin x}}{{4\sqrt{1-(\cos x)^4}}} ) with respect to ( x ), you can use the substitution ( u = \cos x ). Then, ( du = -\sin x , dx ). This substitution will simplify the integral. After substitution, you'll get an integral in terms of ( u ) which can be more easily evaluated. The resulting integral will be a rational function that can be integrated using standard techniques such as partial fractions or trigonometric identities. Once you integrate with respect to ( u ), don't forget to convert back to the original variable ( x ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.