How do you evaluate the definite integral #int (2x^3)dx# from [1,3]?

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Answer 1 · 2016-12-11T10:03:38

#int_1^3 2x^3dx = 40#

First, we will consider the integral as I with the limits:

#I = int_1^3 2x^3dx#

We will take out the constant as it is in multiplication with the variable:

#I = 2 int_1^3 x^3dx#

We know the power rule of integration:

#I = int x^ndx = (x^(n+1))/(n+1)#

Applying power rule on the integral:

#I = 2 [(x^(3+1))/(3+1)]_1^3#

#I = 2 [(x^4)/4]_1^3#

Now we can put the limits of the integration, we know the rule of limits of integration:

Upper Limit-Lower Limit

Hence:

#I = 2 [((3^4)/4)-(1^4/4)]#

We know, #3^4=81# and #1^4=1#

#I = 2 [((81)/4)-(1/4)]#

As the base is same, we can directly subtract #81# and #1#

#I = 2 [((81-1)/4)]#

#I = 2 [80/4]#

We can divide #80# by #4#

#I = 2.20#

#I=40#

Hence:

#int_1^3 2x^3dx = 40#