SATHEE: Indefinite Integration 2 Question 5

Indefinite Integration 2 Question 5

5. Given a function $f (x)$ such that it is integrable over every interval on the real line and $f (t + x) = f (x)$ , for every $x$ and a real $t$ , then show that the integral $\int_{a}^{a + t} f (x) d x$ is independent of $a$ .

$(1984, 4$ M)

Integer Answer Type Question

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Solution:

Let $φ (a) = \int_{a}^{a + t} f (x) d x$

On differentiating w.r.t. $a$ , we get

$φ^{'} (a) = f (a + t) \cdot 1 - f (a) \cdot 1 = 0$ [given, $f (x + t) = f (x)$ ]

$∴ φ (a)$ is constant.

$\Rightarrow \int_{a}^{a + t} f (x) d x$ is independent of $a$ .

$f (x)$ and $\cos π x$ both are periodic with period 2 and both are even.

$∴ \int_{- 10}^{10} f (x) \cos π x d x = 2 \int_{0}^{10} f (x) \cos π x d x$

$= 10 \int_{0}^{2} f (x) \cos π x d x$

Now, $\int_{0}^{1} f (x) \cos π x d x$

$= \int_{0}^{1} (1 - x) \cos π x d x = - \int_{0}^{1} u \cos π u d u$

and $\int_{1}^{2} f (x) \cos π x d x = \int_{1}^{2} (x - 1) \cos π x d x$

$= - \int_{0}^{1} u \cos π u d u$

$∴ \int_{- 10}^{10} f (x) \cos π x d x = - 20 \int_{0}^{1} u \cos π u d u = \frac{40}{π^{2}}$

$\Rightarrow \frac{π^{2}}{10} \int_{- 10}^{10} f (x) \cos π x d x = 4$

Indefinite Integration 2 Question 5

5. Given a function $f (x)$ such that it is integrable over every interval on the real line and $f (t + x) = f (x)$ , for every $x$ and a real $t$ , then show that the integral $\int_{a}^{a + t} f (x) d x$ is independent of $a$ .

Integer Answer Type Question

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Indefinite Integration 2 Question 5

5. Given a function f(x) such that it is integrable over every interval on the real line and f(t+x)=f(x), for every x and a real t, then show that the integral ∫aa+tf(x)dx is independent of a.

Integer Answer Type Question

Solution:

Engineering Preparation

Medical Preparation

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5. Given a function $f (x)$ such that it is integrable over every interval on the real line and $f (t + x) = f (x)$ , for every $x$ and a real $t$ , then show that the integral $\int_{a}^{a + t} f (x) d x$ is independent of $a$ .