Let f x be a given integrable function such that fx-k-fx for all x- R- Then - a a-k f x dx depends for its value on

If a function $y = f (x)$ is such that $f^{'} (x)$ is continuous function and satisfies $(f (x))^{2} = K + \int_{0}^{x} ((f (t))^{2} + (f^{'} (t))^{2}) d t, K ϵ R^{+}$ , then

Q. Let the functions f and g be defined by

f (x) = (x - 3)

and

g (x) = \int_{k, w h e n x = - 3}^{\frac{x^{2} - 9}{x + 3}, w h e n x \neq - 3}

find the value of k such that f(x) = g(x) for all

x \in R

Q. Let

f : R \to A = {y : 0 \leq y < \frac{π}{2}}

be a function such that

f (x) = {tan}^{- 1} (x^{2} + x + k),

where

k

is a constant. The value of

k

for which

f

is an onto function is

Q. Let

f (x)

be a continuous function such that

f (x)

does not vanish for all

x ϵ R .

\int_{2}^{3} f (x) d x = \int_{- 2}^{3}

f (x) d x

then

x ϵ R,

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Let f(x) be a given integrable function such that f(x+k)=f(x) for all xϵR. Then ∫a+kaf(x)dx depends for its value on

The correct option is B k onlyI=∫a+kaf(x)dxSubstitute x=t+a⇒dx=dtWe getI=∫k0f(t+k)dt=∫k0f(x+k)dx=∫k0f(x)dx=[F(x)]k0=F(k)−F(0)

Let $f (x)$ be a given integrable function such that $f (x + k) = f (x)$ for all $x ϵ R$ . Then $\int_{a}^{a + k} f (x) d x$ depends for its value on

The correct option is B $k$ only
$I = \int_{a}^{a + k} f (x) d x$
Substitute $x = t + a \Rightarrow d x = d t$
We get
$I = \int_{0}^{k} f (t + k) d t = \int_{0}^{k} f (x + k) d x = \int_{0}^{k} f (x) d x = {[F (x)]}_{0}^{k} = F (k) - F (0)$