Evaluate the Definite Integral from 1 to 3 of (5+1/x+e^x)dx

Question

jkristia · Answer

What is the antiderivative of 5, 1/x and e^x ?
When you have that, call it F(x), then find the value F(3) - F(1)

anonymous · Answer

antiderivative of 5 would be 5x and for 1/x it would be x^-1 (?) and e^x would just be e^x

anonymous · Answer

actually 1/x's antiderivative would be ln x

anonymous · Answer

CAn anyone help?

e.mccormick · Answer

Yah,
$$\int_1^3\left(5+\frac{1}{x}+e^x\right)dx= \left( 5x+\ln x + e^x\right)|_1^3$$

e.mccormick · Answer

So the next step is to evaluate them on that range.  Did you understand what jkristia meant by F(3)-F(1)?

e.mccormick · Answer

If:
$$f(x)=5+\frac{1}{x}+e^x $$then the antiderivative
$$F(x)=\left( 5x+\ln x + e^x+C\right)$$
When evaluating over a range of:
$$\int_a^b$$
You solve for $F(b)-F(a)$, which should have been shown to you with limits, and because no matter what F(x) is, this will invove:
polynomial + C - [ polynomial + C] Which can be rewritten as:
polynomial - polynomial + C - C 
And since C-C=0, you can ignore the C.

That means in your problem you evaluate:
$F(3)-F(1)$ where you use the C-less form of $F(x)=\left( 5x+\ln x + e^x\right)$