First slide

Evaluation of definite integrals

Question

The equation of the tangent to the curve
$y = \int_{x}^{x^{2}} \log t d t at x = 2$ is

Difficult

A

$y - 6 \log 2 = 7 \log 2 (x - 2)$
B

$y - \log 2 e^{1 / 3} = (\log 2) x$
C

$y - 8 \log 2 e^{- 1 / 3} = 5 \log 2 x$
D

$y + 8 \log 2 + 2 = (7 \log 2) x$

Solution

${y = \int_{x}^{x^{2}} \log t d t = (t \log t - t)]}_{x}^{x^{2}}$
$\begin{aligned} = (2 x^{2} - x) \log x - x^{2} + x \\ \Rightarrow y (2) = 6 \log 2 - 2 \end{aligned}$
Also, $y^{'} (2) = 7 \log 2$
Thus, equation of tangent at $x = 2$ is
$\begin{array}{l} y - (6 \log 2 - 2) = (7 \log 2) (x - 2) \\ or y + 8 \log 2 + 2 = (7 \log 2) (x) \end{array}$

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