Question

a
∫ x/2+x⁸ dx ,Evaluate it.
-a

Answer 1

Answer:

2a⁹/9

Step-by-step explanation:

a

∫ x/2+x⁸ dx

-a

on integrating wrt x we get

a

∫  (x²/4) +x⁹/9

-a

= a²/4 + a⁹/9 - (-a)²/4 - (-a)⁹/9

= a²/4 + a⁹/9  - a²/4 + a⁹/9

= 2a⁹/9

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO EVALUATE

$\displaystyle \sf{ \int\limits_{- a}^{a} \frac{x}{2 + {x}^{8} } \: \, dx }$

FORMULA TO BE IMPLEMENTED

1. A function is said to be odd function if f(-x) = - f(x)

2. If f(x) is a odd function then

$\displaystyle \sf{ \int\limits_{- a}^{a} f(x) \: \, dx } = 0$

EVALUATION

Here

$\displaystyle \sf{f(x) = \: \frac{x}{2 + {x}^{8} } }$

So

$\displaystyle \sf{f( - x) = \: \frac{ - x}{2 + {( - x )\: }^{8} } }$

$\implies \: \displaystyle \sf{f( - x) = \: \frac{ - x}{2 + {x}^{8} } }$

$\implies \: \displaystyle \sf{f( - x) = \: - f(x) }$

Hence f(x) is a odd function

Hence by the above mentioned formula of Definite Integral we get

$\displaystyle \sf{ \int\limits_{- a}^{a} f(x) \: \, dx } = 0$

Hence

$\displaystyle \sf{ \int\limits_{- a}^{a} \frac{x}{2 + {x}^{8} } \: \, dx } = 0$

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