First slide

Evaluation of definite integrals

Question

Suppose f is such that f(-x) = -f(x) for every real x and $\int_{0}^{1} f (x) dx = 5$ then $\int_{- 1}^{0} f (t) dt$ is equal to

Moderate

A

10
B

5
C

0
D

-5

Solution

Given $f (- x) = - f (x), \forall$ values of real x.
We know that,
$\begin{array}{l} \int_{- a}^{a} f (x) dx = 0 = \int_{- a}^{0} f (x) dx + \int_{0}^{a} f (x) dx \\ \Rightarrow \int_{- 1}^{0} f (x) dx + \int_{0}^{1} f (x) dx = 0 \\ \Rightarrow \int_{- 1}^{0} f (x) dx = - 5 [∵ \int_{0}^{1} f (x) dx = 5] \\ \Rightarrow \int_{- 1}^{0} f (t) dt = - 5 \end{array}$

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