# How do you integrate #f(x)=e^xsinxcosx# using the product rule?

##### 1 Answer

# f'(x) = e^x(sinxcosx + cos^2x - sin^2x) #

#### Explanation:

If you are studying maths, then you should learn the Product Rule for Differentiation, and practice how to use it:

# d/dx(uv)=u(dv)/dx+(du)/dxv # , or,# (uv)' = (du)v + u(dv) #

I was taught to remember the rule in words; "*The first times the derivative of the second plus the derivative of the first times the second* ".

This can be extended to three products:

# d/dx(uvw)=uv(dw)/dx+u(dv)/dxw + (du)/dxvw#

So with

Applying the product rule we get:

# d/dx(uvw) = (du)/dxvw + u(dv)/dxw+ + uv(dw)/dx#

# :. f'(x) = (e^x)(sinx)(cosx) + (e^x)(cosx)(cosx) + (e^x)(sinx)(-sinx) #

# " " = e^x(sinxcosx + cos^2x - sin^2x) #