AP rove that -02 a f x dx -2 -0 a f X dx- if f 2 a - x - f x and evaluate -02 -cos 5 x dxif f 2 a - x - f x b Find the values of a and b such that the function defined by f x - left begin matrix 5- - if x - ax - b - if - 2- x -10 - is a - continous function 21 - - if x geq 10 lendmatrix right-

(a) ∫_0^2a f(x) dx = ∫_0^a f(x) dx + ∫_a^2a f(x) dx For derivations Let I_1 = ∫_0^2a f(x) dx For Problem Put 2a x = t ⇒ x = 2a t dx= dt x = a ...

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Polynomial Functions

aP rove that ...

Question

$(a) P r o v e t h a t \int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (X) d x, i f f (2 a - x) = f (x) a n d e v a l u a t e \int_{0}^{2 π} c o s^{5} x d x$
$i f f (2 a - x) = - f (x)$

(b) Find the values of a and b such that the function defined by

$f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 5, & i f x \leq 2 a x + b & i f 2 < x < 10 i s a c o n t i n o u s f u n c t i o n 21, & i f x \geq 10 \end{matrix}$

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a

b

f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 5, & i f & x \leq 2 a x + b, & i f & 2 < x < 10 21, & i f & x \geq 10 \end{matrix}

f (2 a - x) = f (x)

\int_{0}^{a} f (x) d x = λ

\int_{0}^{2 a} f (x) d x

\int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x

a

b

f (x) =

⎧ ⎨ ⎩ \begin{matrix} 5, i f x \leq 2 a x + b, i f 2 < x < 10 21, i f x \geq 10 \end{matrix}

\int_{0}^{π} x f (sin x) d x

\frac{π}{2} \int_{0}^{π} f (sin x) d x

π \int_{0}^{π / 2} f (sin x) d x

$\int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x$ and $\int_{0}^{2 a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ If $f (2 a - x) = f (x)$

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