SATHEE: Indefinite Integration 1 Question 6

Indefinite Integration 1 Question 6

7. Let $f (x) = \int_{0}^{x} g (t) d t$ , where $g$ is a non-zero even function. If $f (x + 5) = g (x)$ , then $\int_{0}^{x} f (t) d t$ equals

(2019 Main, 8 April II)

(a) $5 \int_{x + 5}^{5} g (t) d t$

(b) $\int_{5}^{x + 5} g (t) d t$

(d) $\int_{x + 5}^{5} g (t) d t$

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Solution:

Given, $f (x) = \int_{0}^{x} g (t) d t$

On replacing $x$ by $(- x)$ , we get

$f (- x) = \int_{0}^{- x} g (t) d t$

Now, put $t = - u$ , so

$f (- x) = - \int_{0}^{x} g (- u) d u = - \int_{0}^{x} g (u) d u = - f (x)$

$[∵ g$ is an even function] $\Rightarrow f (- x) = - f (x) \Rightarrow f$ is an odd function.

Now, it is given that $f (x + 5) = g (x)$

$\begin{aligned} ∴ f (5 - x) = g (- x) = g (x) = f (x + 5) \\ \Rightarrow f (5 - x) = f (x + 5) \end{aligned}$

Put $t = u + 5 \Rightarrow t - 5 = u \Rightarrow d t = d u$

$\begin{array}{ll} ∴ & I = \int_{- 5}^{x - 5} f (u + 5) d u = \int_{- 5}^{x - 5} g (u) d u \\ Put u = - t \Rightarrow d u = - d t, we get \\ I = - \int_{5}^{5 - x} g (- t) d t = \int_{5 - x}^{5} g (t) d t \end{array}$

$\begin{aligned} [∵ - \int_{a}^{b} f (x) d x = \int_{b}^{a} f (x) d x and g is an even function] \\ I = \int_{- x}^{5} f^{'} (t) d t [by Leibnitz rule f^{'} (x) = g (x)] \\ = f (5) - f (5 - x) = f (5) - f (5 + x) [from Eq. (i)] \\ = \int_{5 + x}^{5} f^{'} (t) d t = \int_{5 + x}^{5} g (t) d t \end{aligned}$

Indefinite Integration 1 Question 6

7. Let $f (x) = \int_{0}^{x} g (t) d t$ , where $g$ is a non-zero even function. If $f (x + 5) = g (x)$ , then $\int_{0}^{x} f (t) d t$ equals

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Indefinite Integration 1 Question 6

7. Let f(x)=∫0xg(t)dt, where g is a non-zero even function. If f(x+5)=g(x), then ∫0xf(t)dt equals

Solution:

Engineering Preparation

Medical Preparation

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Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

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NCERT Exercise Video Solution

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7. Let $f (x) = \int_{0}^{x} g (t) d t$ , where $g$ is a non-zero even function. If $f (x + 5) = g (x)$ , then $\int_{0}^{x} f (t) d t$ equals