SATHEE: Indefinite Integration 3 Question 21

Indefinite Integration 3 Question 21

21. If ’ $f$ ’ is a continuous function with $\int_{0}^{x} f (t) d t \to \infty$ as $| x | \to \infty$ , then show that every line $y = m x$ intersects the curve $y^{2} + \int_{0}^{x} f (t) d t = 2$

$(1991, 2 M)$

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Solution:

Since, $f$ is continuous function and $\int_{0}^{x} f (t) d t \to \infty$ , as $| x | \to \infty$ . To show that every line $y = m x$ intersects the curve $y^{2} + \int_{0}^{x} f (t) d t = 2$

At $x = 0, y = \pm \sqrt{2}$

Hence, $(0, \sqrt{2}), (0, - \sqrt{2})$ are the point of intersection of the curve with the $Y$ -axis. As $x \to \infty, \int_{0}^{x} f (t) d t \to \infty$ for a particular $x$ (say $x_{n}$ ), then $\int_{0}^{x} f (t) d t = 2$ and for this value of $x, y = 0$

The curve is symmetrical about $X$ -axis.

Thus, we have that there must be some $x$ , such that $f (x_{n}) = 2$ .

Thus, $y = m x$ intersects this closed curve for all values of $m$ .

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Indefinite Integration 3 Question 21

21. If ’ f ’ is a continuous function with ∫0xf(t)dt→∞ as |x|→∞, then show that every line y=mx intersects the curve y2+∫0xf(t)dt=2

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21. If ’ $f$ ’ is a continuous function with $\int_{0}^{x} f (t) d t \to \infty$ as $| x | \to \infty$ , then show that every line $y = m x$ intersects the curve $y^{2} + \int_{0}^{x} f (t) d t = 2$