How do you find the integral of #int sin^6(x) cos^3(x) dx#?

Answer 1

#int sin^mxcos^nx dx# with one (or both) #m, n# odd should be added to your mathematics recipe book.

We can integrate by substitution since sine and cosine are derivatives of one another (up to a mminus sign).

Regroup one the the functions that has an odd exponent to join #dx#. This pair will be #du# when we make a substitution. It also leaves an even power that can be rewritten using #sin^2x+cos^2x =1#

Complete the substitution, then integrate, expand, and resultant polynomial.

To complete the substitution, reverse it.

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 ( \int \sin^6(x) \cos^3(x) , dx ), you can use the method of integration by parts. Integration by parts states that ( \int u , dv = uv - \int v , du ), where ( u ) and ( v ) are differentiable functions of ( x ).

In this case, let ( u = \sin^6(x) ) and ( dv = \cos^3(x) , dx ). Then, differentiate ( u ) to find ( du ), and integrate ( dv ) to find ( v ).

Differentiating ( u ): [ du = 6\sin^5(x) \cos(x) , dx ]

Integrating ( dv ): [ v = \int \cos^3(x) , dx ] [ = \int \cos^2(x) \cos(x) , dx ] [ = \int (1 - \sin^2(x)) \cos(x) , dx ] [ = \int (\cos(x) - \sin^2(x) \cos(x)) , dx ] [ = \sin(x) - \int \sin^2(x) \cos(x) , dx ]

Now, we have ( u ), ( du ), ( v ), and ( dv ), so we can apply integration by parts:

[ \int \sin^6(x) \cos^3(x) , dx = uv - \int v , du ] [ = \sin^6(x) \sin(x) - \int (\sin(x) - \int \sin^2(x) \cos(x) , dx) (6\sin^5(x) \cos(x)) , dx ]

This integral is now reduced to the integral of ( \sin^2(x) \cos(x) ), which can be solved using integration by parts again, or through trigonometric identities.

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