What is the integral of #cos^2[x] sin[x]#?

Answer 1

#-cos^3(x)/3+C#

We want to find:

#intcos^2(x)sin(x)dx#

Remember that the derivative of sine and cosine are basically one another. Since we only have one sine function, it will serve very well as the derivative of the cosine function when we substitute.

Let #u=cos(x)#. This implies that #du=-sin(x)dx#. Note that #sin(x)dx=-du#, which is what we have here. Also note that #cos^2(x)=u^2#.

Thus:

#intunderbrace(cos^2(x))_(u^2)overbrace(sin(x)dx)^(-du)=-intu^2du#
Then using the common rule: #intu^ndu=u^(n+1)/(n+1)+C#, where #n!=-1#, we see that:
#-intu^2du=-u^3/3+C=-cos^3(x)/3+C#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the integral of ( \cos^2(x) \sin(x) ), you can use the substitution method. Let ( u = \cos(x) ), then ( du = -\sin(x) dx ).

So, the integral becomes:

[ \int \cos^2(x) \sin(x) dx = -\int u^2 du ]

Now, integrate ( -\int u^2 du ):

[ -\int u^2 du = -\frac{u^3}{3} + C ]

Finally, replace ( u ) with ( \cos(x) ):

[ \int \cos^2(x) \sin(x) dx = -\frac{\cos^3(x)}{3} + C ]

Where ( C ) is the constant of integration.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

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.

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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7