Let p be a polynomial with real coefficients such that ∫ 0 1 p t d t = 0 . Which of the following statements is always true?

p has a root in the interval [ 0 , 1 ]

All roots of p lie in the interval [ 0 , 1 ]

p has no roots in the interval [ 0 , 1 ] .

p has exactly one root in the interval [ 0 , 1 ] .

Let p be a polynomial with real coefficients such that ∫ 0 1 p t d t = 0 . Which of the following statements is always true?

p has a root in the interval [ 0 , 1 ]

Let, n ≥ 2 be a natural number and 0 < θ < π 2 . Then ∫ s i n n θ - s i n θ 1 n c o s θ s i n n + 1 θ d θ , is equal to

n n 2 - 1 1 - 1 s i n n - 1 θ n + 1 n + c

n n 2 - 1 1 - 1 s i n n + 1 θ n + 1 n + c

n n 2 + 1 1 - 1 s i n n - 1 θ n + 1 n + c

n n 2 - 1 1 + 1 s i n n - 1 θ n + 1 n + c

If ∫ f ( x ) d x = ψ ( x ) , then ∫ x 5 f ( x 3 ) d x , is equal to

1 3 x 3 ψ ( x 3 ) − ∫ x 2 ψ ( x 3 ) d x + c

1 3 [ x 3 ψ ( x 3 ) − ∫ x 3 ψ ( x 3 ) d x ] + c

1 3 [ x 3 ψ ( x 3 ) − ∫ x 2 ψ ( x 3 ) d x ] + c

1 3 x 3 ψ ( x 3 ) − 3 ∫ x 3 ψ ( x 3 ) d x + c

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Let $p$ be a polynomial with real coefficients such that $\int_{0}^{1} p (t) d t = 0$ . Which of the following statements is always true?

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Important Questions on Definite Integration

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

The number of continuous function

f : [0,1] \to [0,1]

such that

f (x) < x^{2}

for all

x

and

\int_{0}^{1} f (x) d x = \frac{1}{3}

is:

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

\int \frac{\log (t + \sqrt{1 + t^{2}})}{\sqrt{1 + t^{2}}} d t = \frac{1}{2} {(g (t))}^{2} + c

, where

c

is a constant, then

g (2)

, is equal to

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

The number of continuous functions

f : [0, 1] \to R

that satisfy

\int_{0}^{1} x f (x) d x = \frac{1}{3} + \frac{1}{4} \int_{0}^{1} {(f (x))}^{2} d x

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

\int \frac{\tan x}{1 + \tan x + \tan^{2} x} d x = x - \frac{K}{\sqrt{A}} \tan^{- 1} (\frac{K \tan x + 1}{\sqrt{A}}) + C

, (

C

is a constant of integration), then the ordered pair

(K, A)

is equal to

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

\int x^{5} e^{- 4 x^{3}} d x = \frac{1}{48} e^{- 4 x^{3}} f (x) + C

, where

C

is a constant of integration, then

f (x)

is equal to

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

Let,

I_{n} = \int \tan^{n} x d x (n > 1)

. If

I_{4} + I_{6} = a \tan^{5} x + b x^{5} + c

, then the ordered pair

(a, b)

, is equal to

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

The integral

\int \frac{d x}{{(x + 1)}^{\frac{3}{4}} {(x - 2)}^{\frac{5}{4}}}

, is equal to

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

f (\frac{3 x - 4}{3 x + 4}) = x + 2, x \neq - \frac{4}{3}

, and

\int f (x) d x = A \log |1 - x| + B x + C

, then the ordered pair

(A, B)

is equal to

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

The integral

\int \frac{3 x^{13} + 2 x^{11}}{{(2 x^{4} + 3 x^{2} + 1)}^{4}} d x

, is equal to

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

f (x) = \int \frac{(5 x^{8} + 7 x^{6})}{{(x^{2} + 1 + 2 x^{7})}^{2}} d x

(x \geq 0),

and

f (0) = 0,

then the value of

f (1)

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

\int \frac{\sec^{8} x}{cosec x} d x =

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

Let,

n \geq 2

be a natural number and

0 < θ < \frac{π}{2} .

Then

\int \frac{{(s i n^{n} θ - s i n θ)}^{\frac{1}{n}} c o s θ}{s i n^{n + 1} θ} d θ,

is equal to

EASY

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

The value of

\int_{0}^{2} \frac{d x}{x^{2} - 2 x + 2}

is equal to

EASY

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

The integral

\int_{\frac{π}{4}}^{\frac{3 π}{4}} \frac{d x}{1 + \cos x}

is equal to

EASY

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

The integral

\int_{\frac{π}{12}}^{\frac{π}{4}} \frac{8 \cos 2 x}{{(\tan x + \cot x)}^{3}} d x

equals

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

Suppose that the earth is a sphere of radius

6400

kilometers. The height from the earth's surface from where exactly a fourth of the earth's surface is visible, is

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

For

x, t \in R l e t p_{t} (x) = (\sin t) x^{2} - (2 \cos t) x + \sin t

be a family of quadratic polynomials in

x

with variable coefficients. Let

A (t) = \int_{0}^{1} p_{t} (x) d x

. Which of the following statement are true?

(I)

A (t) < 0

for all

t

.

(II)

A (t)

has infinitely many critical points.

(III)

A (t) = 0

for infinitely many

t

.

(IV)

A' (t) < 0

for all

t

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

\int f (x) d x = ψ (x)

, then

\int x^{5} f (x^{3}) d x

, is equal to

HARD

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

The value of

\int_{- π / 2}^{π / 2} \frac{d x}{[x] + [\sin x] + 4},

where

[t]

denotes the greatest integer less than or equal to

t,

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Basics of Definite Integral

The integral $\int \frac{d x}{x^{2} {(x^{4} + 1)}^{\frac{3}{4}}}$ equals to

Let p be a polynomial with real coefficients such that ∫01ptdt=0. Which of the following statements is always true?

Important Questions on Definite Integration

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Let $p$ be a polynomial with real coefficients such that $\int_{0}^{1} p (t) d t = 0$ . Which of the following statements is always true?