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Let ∫_a^bx f(x)dx ........(i) Then, by a property of definite integrals I=∫_a^b(a+b x)f(a+b x)dx=∫_a^b(a+b x)f(x) dx ...( ...

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Standard XII

Mathematics

Area between x=g(y) and y Axis

Choose the co...

Question

Choose the correct answer.

$I f f (a + b - x) = f (x), t h e n \int_{a}^{b} x f (x) d x i s e q u a l t o (a) \frac{a + b}{2} \int_{a}^{b} f (b - x) d x (b) \frac{a + b}{2} \int_{a}^{b} f (b + x) d x (c) \frac{b - a}{2} \int_{a}^{b} f (x) d x (d) \frac{a + b}{2} \int_{a}^{b} f (x) d x$

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Solution

Let $\int_{a}^{b} x f (x) d x$ ........(i)

Then, by a property of definite integrals

$I = \int_{a}^{b} (a + b - x) f (a + b - x) d x = \int_{a}^{b} (a + b - x) f (x) d x . . . (i i) (∵ G i v e n f (a + b - x) = f (x))$

On adding Eqs. (i) and (ii), we get

$2 I = \int_{a}^{b} (a + b) f (x) d x \Rightarrow I = \frac{a + b}{2} \int_{a}^{b} f (x) d x$

$∴$ Hence, the option (d) is correct.

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Similar questions

Which of the following is correct?
a) $(a - b)^{2} = a^{2} + 2 a b - b^{2}$
b) $(a - b)^{2} = a^{2} - 2 a b + b^{2}$
c) $(a - b)^{2} = a^{2} - b^{2}$
d) $(a + b)^{2} = a^{2} + 2 a b - b^{2}$

Q. The correct order of stability for the following species is:

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Q. Choose the correct option for the given equation.

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\frac{\sqrt{a^{2} - b^{2}} + a}{\sqrt{a^{2} + b^{2}} + b} \div \frac{\sqrt{a^{2} + b^{2}} - b}{a - \sqrt{a^{2} - b^{2}}}

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\frac{(a^{2} + b^{2})}{(a^{2} - b^{2})}

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Choose the correct answer. If f(a+b−x)=f(x), then ∫bax f(x) dx is equal to(a)a+b2∫baf(b−x)dx(b)a+b2∫baf(b+x)dx(c)b−a2∫baf(x) dx(d)a+b2∫baf(x) dx

Let ∫bax f(x)dx ........(i) Then, by a property of definite integrals I=∫ba(a+b−x)f(a+b−x)dx=∫ba(a+b−x)f(x) dx ...(ii) (∵ Given f(a+b−x)=f(x)) On adding Eqs. (i) and (ii), we get 2I=∫ba(a+b)f(x) dx ⇒ I=a+b2∫baf(x) dx ∴ Hence, the option (d) is correct.

Choose the correct answer.

$I f f (a + b - x) = f (x), t h e n \int_{a}^{b} x f (x) d x i s e q u a l t o (a) \frac{a + b}{2} \int_{a}^{b} f (b - x) d x (b) \frac{a + b}{2} \int_{a}^{b} f (b + x) d x (c) \frac{b - a}{2} \int_{a}^{b} f (x) d x (d) \frac{a + b}{2} \int_{a}^{b} f (x) d x$