- 02 afx dx is equal toa 2 - 0 afx dxb 0c - 0 afx d x- 0 a f 2 a-x d xd - 0 a f x d x- 02 a f 2 a-x d x

C #160; #8747;0afx #160;dx+ #8747;0af2a x #160;dx According #160;to #160;the #160;additivity #160;property #160;of #160;integrals, #874 ...

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Byju's Answer

Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 2

∫ 02 afx dx i...

Question

$\int_{0}^{2 a} f (x) d x$ is equal to

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

(b) 0

(c) $\int_{0}^{a} f (x) d x + \int_{0}^{a} f (2 a - x) d x$

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

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Solution

$(c) \int_{0}^{a} f (x) d x + \int_{0}^{a} f (2 a - x) d x$

$A c c o r d i n g t o t h e a d d i t i v i t y p r o p e r t y o f i n t e g r a l s, \int_{a}^{b} f (x) d x = \int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x, w h e r e a < c < b u \sin g t h i s p r o p e r t y, \int_{0}^{2 a} f (x) d x = \int_{0}^{a} f (x) d x + \int_{0}^{2 a} f (x) d x . . . . . . (1) N o w, c o n s i d e r t h e i n t e g r a l, \int_{0}^{2 a} f (x) d x L e t x = 2 a - t . T h e n d x = d (2 a - t) \Rightarrow d x = - d t A l s o, x = a \Rightarrow t = a a n d x = 2 a \Rightarrow t = 0 T h e r e f o r e, \int_{a}^{2 a} f (x) d x = - \int_{a}^{0} f (2 a - t) d t \Rightarrow \int_{a}^{2 a} f (x) d x = \int_{0}^{a} f (2 a - t) d t \Rightarrow \int_{a}^{2 a} f (x) d x = \int_{0}^{a} f (2 a - x) d x S u b s t i t u t i n g t h i s i n e q u a t i o n (1) w e g e t, \int_{0}^{2 a} f (x) d x = \int_{0}^{a} f (x) d x + \int_{0}^{a} f (2 a - x) d x$ →

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Similar questions

Q. Prove that:

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

and hence evaluate

\int_{0}^{1} x^{2} {(1 - x)}^{n} d x

Q. If f(2a − x) = −f(x), prove that

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

Q. If f is an integrable function, show that
(i)

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

(ii)

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

Q. Prove that :

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

If both $lim x \to a f (x)$ and $lim x \to a g (x)$ and exist finitely and $lim x \to a g (x) = 0$ , then
$lim x \to a \frac{f (x)}{g (x)} = \frac{lim x \to a f (x)}{lim x \to a g (x)}$

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∫02afx dx is equal to (a) 2∫0afx dx (b) 0 (c) ∫0afx dx+∫0af2a-x dx (d) ∫0afx dx+∫02af2a-x dx

$\int_{0}^{2 a} f (x) d x$ is equal to

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

(b) 0

(c) $\int_{0}^{a} f (x) d x + \int_{0}^{a} f (2 a - x) d x$

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