The Fundamental Theorem of Calculus

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The discussion revolves around evaluating the definite integral of the function (3/x^2 - 1) from 1 to 2, with the correct answer being 1/2. The user initially struggles with finding the antiderivative, mistakenly applying the power rule. After realizing the error in exponent manipulation, they correctly derive the function and confirm the result. The importance of verifying answers through differentiation is emphasized. Ultimately, the user successfully arrives at the correct answer of 0.5.
thegoosegirl42
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Homework Statement


Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result.
Integral from [1 to 2] of (3/x^2 - 1)

Homework Equations


The answer is 1/2
f(x)dx= F(b) - F(a)

The Attempt at a Solution


I tried taking it to make it -x^-3 - 1x as the antiderivative but when I put 2 in and then 1 I don't get the answer.
 
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thegoosegirl42 said:
I tried taking it to make it -x^-3 - 1x as the antiderivative
Take the derivative to check your answer. Is it correct?
 
Nathanael said:
Take the derivative to check your answer. Is it correct?
Gosh dang it I was going down in exponent when I should have been going up. The new derivative would be -3x^-1 -1x making the answer be -3.5+4 Thus the answer is .5. Thank you so much.
 

Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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