How do you evaluate the definite integral #int (2x) dx# from #[2,3]#?

1 Answer
Sep 12, 2016

#5#

Explanation:

in terms of the indefinite integral, use the Power Rule for integration:

#int x^n \ dx = (x^(n+1))/(n+1) + C#

And, with constant #alpha#:

#int alpha x^n \ dx = (alpha \ x^(n+1))/(n+1) + C#

Or, if you like, lift the constant outside the integration:

#int alpha x^n \ dx = alpha int x^n \ dx#

#=alpha ( \ x^(n+1))/(n+1) + C = (alpha \ x^(n+1))/(n+1) + C#

I'm labouring this, deliberately.

So

#int 2 x \ dx#

# = 2 int x^color(red)(1) \ dx#

from the Power Rule
# =2 ( x^(1+1))/(1+1) + C#

# =x^2 + C qquad triangle#

Finally, if in doubt, differentiate your result in #triangle#, because differentiation and integration are like inverse processes

# d/dx (x^2 + C) = d/dx (x^2) + d/dx(C) = 2x + 0 = 2x# Voila!!

Now for the definite integral

#int_2^3 2 x \ dx#

#= 2 int_2^3 x \ dx#

from the Power Rule
#= 2 [ x^2/2 ]_2^3#

#= [ x^2 ]_2^3#

#= 9 - 4 = 5#