Question

Use Part 2 of the fundamental Theorem of Calculus to find the derivatives. \frac{d}{dx}\int_{1}^{x}\ln t dt

Derivatives
ANSWERED
asked 2021-05-02
Use Part 2 of the fundamental Theorem of Calculus to find the derivatives.
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{1}}}^{{{x}}}}{\ln{{t}}}{\left.{d}{t}\right.}\)

Answers (1)

2021-05-03
Step 1
Given
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{1}}}^{{{x}}}}{\ln{{t}}}{\left.{d}{t}\right.}\)
Step 2
Solution
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{1}}}^{{{x}}}}{\ln{{t}}}{\left.{d}{t}\right.}\)
\(\displaystyle={\ln{{x}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{x}-{\ln{{1}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{1}\)
\(\displaystyle={\ln{{x}}}-{0}\)
\(\displaystyle={\ln{{x}}}\)
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