SOLUTION: Find the area under the curve: y = 4x + (1/x2) from x = 1 to x = 4

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Question 369761: Find the area under the curve:
y = 4x + (1/x2) from x = 1 to x = 4

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Assuming the equation is
y+=+4x+%2B+%281%2Fx%5E2%29 (Please use "^" to indicate exponents.)

The answer will the the value of the definite integral:
int%284x+%2B+1%2Fx%5E2%2C+dx%2C+1%2C+4%29
Since integrals of powers of x are simple, I am going to rewrite the second term as the appropriate power of x:
int%284x+%2B+x%5E%28-2%29%2C+dx%2C+1%2C+4%29
Next we can use a property of integrals that tells us the the integral of a sum is the same as the sum of the integrals:
int%284x%2C+dx%2C+1%2C+4%29+%2B+int%28x%5E%28-2%29%2C+dx%2C+1%2C+4%29
Next we can factor out the 4 in the first integral using a property of integrals:
4%2Aint%28x%2C+dx%2C+1%2C+4%29+%2B+int%28x%5E%28-2%29%2C+dx%2C+1%2C+4%29
Each integral is now a power of x. In general int%28x%5En%2C+dx%29+=+x%5E%28n%2B1%29%2F%28n%2B1%29. Using this on both of our integrals we get:
4%28x%5E%281%2B1%29%2F%281%2B1%29%29+%2B+x%5E%28-2%2B1%29%2F%28-2%2B1%29 evaluated from x=1 to x=4
This simplifies as follows:
4%28x%5E2%2F2%29+%2B+x%5E%28-1%29%2F%28-1%29 evaluated from x=1 to x=4
2x%5E2+%2B+%28-x%5E%28-1%29%29 evaluated from x=1 to x=4
Evaluating this expression from 1 to 4:

Simplifying:
%282%2816%29+%2B+%28-1%2F4%29%29+-+%282%2A1+%2B+%28-1%29%29
%2832+%2B+%28-1%2F4%29%29+-+%282+%2B+%28-1%29%29
%28128%2F4+%2B+%28-1%2F4%29%29+-+%281%29
%28127%2F4%29+-+%281%29
%28127%2F4%29+-+%284%2F4%29
123%2F4