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Question: Suppose f is such that f(–x) = –f(x) for every real x and \(\int_{0}^{1}{f(x)dx = 5}\), then \(\int_...

Suppose f is such that f(–x) = –f(x) for every real x and ∫01f(x)dx=5\int_{0}^{1}{f(x)dx = 5}, then ∫−10f(t)dt=\int_{- 1}^{0}{f(t)dt} =

A

10

B

5

C

0

D

–5

Answer

–5

Explanation

Solution

Given, ∫01f(x)dx=5\int_{0}^{1}{f(x)dx = 5}

Put x=−t⇒dx=−dtx = - t \Rightarrow dx = - dt

∴ I=−∫0−1f(−t)dt=−∫−10f(t)dtI = - \int_{0}^{- 1}{f( - t)dt = - \int_{- 1}^{0}{f(t)dt}} ⇒ I=−5I = - 5

(3) ∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx\int_{a}^{b}{f(x)dx = \int_{a}^{c}{f(x)dx} + \int_{c}^{b}{f(x)dx}}, (where a < c < b)

or∫abf(x)dx=∫ac1f(x)dx+∫c1c2f(x)dx+.....+∫cnbf(x)dx;\int_{a}^{b}{f(x)dx} = \int_{a}^{c_{1}}{f(x)dx} + \int_{c_{1}}^{c_{2}}{f(x)dx + ..... + \int_{c_{n}}^{b}{f(x)dx;}} (where a<c1<c2<........cn<ba < c_{1} < c_{2} < ........c_{n} < b)