What is the integral of x(e^x+sin(x^2))?

1 Answer
Mar 10, 2018

The answer is #=(x-1)e^x-1/2cos(x^2)+C#

Explanation:

The integral is

#I=intx(e^x+sin(x^2))dx#

#=intxe^xdx+intxsin(x^2)dx#

#=I_1+I_2#

The first integral is calculated by integration by parts

#intuv'=uv-intuv'#

#u=x#, #=>#, #u'=1#

#v'=e^x#, #=>#, #v=e^x#

Therefore,

#I_1=xe^x-inte^xdx=xe^x-e^x=(x-1)e^x#

The second integral is calculated by substitution

Let #u=x^2#, #=>#, #du=2xdx#

Therefore,

#I_2=intx*(du)/(2x)*sinu#

#=1/2intsinudu#

#=-1/2cosu#

#=-1/2cos(x^2)#

Finally,

#I=I_1+I_2#

#=(x-1)e^x-1/2cos(x^2)+C#