_{-1}¹ \; { x - x²}{3-|x|} d x =

Correct Answer is : 7−18log⁡327 - 18 3/27−218log3

Correct Answer is : 7−18log⁡327 - 18 3/27−218log3

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AP EAPCET 05-Jul-2022 Shift 2 Paper

Section: Mathematics

Question : 77 of 160

Marks: +1, -0

\int\limits_{-1}^{1} \; \frac{\sin x - x^{2}}{3-|x|} d x =

Solution: 👈: Video Solution

Let

I = \int\limits_{-1}^{1} \; \frac{\sin \; x - x^{2}}{3-|x|} dx

= \int\limits_{-1}^{1} \; \frac{\sin \; x}{3-|x|} dx - \int\limits_{-1}^{1} \; \frac{x^{2}}{3-|x|} dx

\; \frac{\sin \; x}{3-|x|}

is an odd function.

\therefore \; \; \int\limits_{-1}^{1} \; \frac{\sin \; x}{3-|x|} dx = 0

\; \frac{x^{2}}{3-|x|}

is an even function,

\therefore \; \; \int\limits_{-1}^{1} \; \frac{\sin \; x}{3-|x|} dx = 0

\; \frac{x^{2}}{3-|x|}

is an even function,

\; \int\limits_{-1}^{1} \; \frac{x^{2}}{3-|x|} dx = 2 \int\limits_{0}^{1} \; \frac{x^{2}}{3-|x|} dx

\; I = -2 \int\limits_{0}^{1} \; \frac{x^{2}}{3-x}

\; \; = 2 \int\limits_{0}^{1} \; \frac{x^{2} - 3x + 3x - 9 + 9}{x-3} dx

\; \; \; = 2 \int\limits_{0}^{1} \left( x + 3 + \; \frac{9}{x-3} \right) dx

\; = 2 \left[ \; \frac{x^{2}}{2} + 3x + 9 \log(x-3) \right]_{0}^{1}

\; = 2 \left[ \frac{1}{2} + 3 + 9 \log(-2) - 9 \log(-3) \right]

\; = 2 \left[ \frac{7}{2} - 9 \log\left( \frac{3}{2} \right) \right] = 7 - 18 \log \frac{3}{2} + C

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