Statement - I : The value of the integral ∫ π / 6 π / 3 d x 1 + tan x is equal to π 6 . Statement - II : ∫ a b f x d x = ∫ a b f a + b - x d x .

Statement - I is false; Statement - II is true.

Statement - I is true; Statement - II is false.

Statement - I true; Statement - II is true; Statement - II is a correct explanation for Statement - I .

Statement - I is true; Statement - II is true; Statement - II is not a correct explanation for Statement - I .

Define g x = ∫ - 3 3 f x - y f y d y , for all real x , where f t = 1 , 0 ≤ t ≤ 1 0 , e l s e w h e r e , then

g x is continuous everywhere and differentiable everywhere except at x = 0 , 1 , 2

g x is not continuous everywhere

g x is continuous everywhere but differentiable nowhere

g x is continuous everywhere and differentiable everywhere except at x = 0, 1

The function f x = sin π x n ! − cos π x n + 1 ! is

periodic, with period 2 n + 1 !

HARD

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If $g (x) = \int_{0}^{x} \cos^{4} t d t,$ then $g (x + π)$ equals

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Important Questions on Functions

MEDIUM

Mathematics>Differential Calculus>Functions>Periodic Functions

The integral

\int_{2}^{4} \frac{log x^{2}}{log x^{2} + log {(6 - x)}^{2}} d x

is equal to

EASY

Mathematics>Differential Calculus>Functions>Periodic Functions

The period of the function

f (x) = \cos 4 x + \tan 3 x

MEDIUM

Mathematics>Differential Calculus>Functions>Periodic Functions

The period of

\sin^{2} θ

MEDIUM

Mathematics>Differential Calculus>Functions>Periodic Functions

Statement - I : The value of the integral $\int_{π / 6}^{π / 3} \frac{d x}{1 + \sqrt{\tan x}}$ is equal to $\frac{π}{6} .$

Statement - II : $\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x .$

HARD

Mathematics>Differential Calculus>Functions>Periodic Functions

a, b

be two fixed positive integers such that

f (a + x) = b + {[b^{3} + 1 - 3 b^{2} f (x) + 3 b {\{f (x)\}}^{2} - {\{f (x)\}}^{3}]}^{1 / 3}

for all real

x

then

f (x)

is a periodic function with period

EASY

Mathematics>Differential Calculus>Functions>Periodic Functions

Let

f

and

g

be continuous functions on

[0, a]

such that

f (x) = f (a - x)

and

g (x) + g (a - x) = 4

, then

\int_{0}^{a} f (x) g (x) d x

is equal to

EASY

Mathematics>Differential Calculus>Functions>Periodic Functions

f (a + x) = f (x)

, then

\int_{0}^{n a} f (x) d x

, where

n \in N

, is equal to

HARD

Mathematics>Differential Calculus>Functions>Periodic Functions

Define

g (x) = \int_{- 3}^{3} f (x - y) f (y) d y,

for all real

x,

where

f (t) = \{\begin{matrix} 1, & 0 \leq t \leq 1 \\ 0, & e l s e w h e r e \end{matrix},

then

EASY

Mathematics>Differential Calculus>Functions>Periodic Functions

f : R \to R

is a function satisfying

f (2 x + 3) + f (2 x + 7) = 2 \forall x \in R

, then the period of

f (x)

EASY

Mathematics>Differential Calculus>Functions>Periodic Functions

The function

f (x) = \tan π x - x + [x]

has period, where

[.]

denotes greatest integer function

HARD

Mathematics>Differential Calculus>Functions>Periodic Functions

Let

f : R \to R

be a function such that

f (2 - x) = f (2 + x)

and

f (4 - x) = f (4 + x)

, for all

x \in R

and

\int_{0}^{2} f (x) d x = 5

. Then the value of

\int_{10}^{50} f (x) d x

EASY

Mathematics>Differential Calculus>Functions>Periodic Functions

Consider the function

f (x) = \cos x^{2}

. Then

HARD

Mathematics>Differential Calculus>Functions>Periodic Functions

The integral

\int_{0}^{π} \sqrt{1 + 4 {sin}^{2} \frac{x}{2} - 4 sin \frac{x}{2}} d x

equals

MEDIUM

Mathematics>Differential Calculus>Functions>Periodic Functions

The value of

\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{x^{2} \cos x}{1 + e^{x}} d x

is equal to

HARD

Mathematics>Differential Calculus>Functions>Periodic Functions

Let

f (x) = \max \{3, x^{2}, \frac{1}{x^{2}}\}

for

\frac{1}{2} \leq x \leq 2 .

Then the value of the integral

\int_{\frac{1}{2}}^{2} f (x) d x

EASY

Mathematics>Differential Calculus>Functions>Periodic Functions

\int_{0}^{3} [x] d x =______,

where

[x]

is greatest integer function

MEDIUM

Mathematics>Differential Calculus>Functions>Periodic Functions

The value of the integral

\int_{- π}^{π} \frac{{c o s}^{2} x}{1 + a^{x}} d x,

a > 0

MEDIUM

Mathematics>Differential Calculus>Functions>Periodic Functions

The function

f (x) = \sin (\frac{π x}{n!}) - \cos (\frac{π x}{(n + 1)!})

EASY

Mathematics>Differential Calculus>Functions>Periodic Functions

Let $f (x) = \sin x + \cos a x$ be a periodic function. Then,

MEDIUM

Mathematics>Differential Calculus>Functions>Periodic Functions

The function $f (x) = |sin 4 x| + |cos 2 x|,$ is a periodic function with a fundamental period

If g(x)= ∫ 0 x cos 4 t dt, then g(x+π) equals

Important Questions on Functions

Learn and Prepare for any exam you want !

If $g (x) = \int_{0}^{x} \cos^{4} t d t,$ then $g (x + π)$ equals