First slide

Evaluation of definite integrals

Question

Let $f : R \to R$ be defined by $f (x) = \int_{0}^{1} \frac{x^{2} + t^{2}}{2 - t} d t$
Then the curve $y = f (x)$ is

Moderate

A

an ellipse
B

a straight line
C

a parabola
D

a hyperbola

Solution

$\begin{array}{l} f (x) = x^{2} \int_{0}^{1} \frac{d t}{2 - t} + \int_{0}^{1} \frac{t^{2}}{2 - t} d t \\ = - x^{2} \log | 2 - t |_{0}^{1} + \int_{0}^{1} (- t - 2 + \frac{4}{2 - t}) d t \\ y = x^{2} \log 2 + (- \frac{1}{2} - 2 + 4 \log 2) \end{array}$
which represent a parabola.

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