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

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

HARD

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Let $\frac{d}{d x} F (x) = (\frac{e^{sin x}}{x}), x > 0$ . If $\int_{1}^{4} \frac{3}{x} e^{sin x^{3}} d x = F (k) - F (1)$ , then one of the possible values of k is

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Important Questions on Integrals

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

\int f (x) d x = ψ (x)

, then

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

, is equal to

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

\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

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

The integral

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

, is equal to

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

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>Integrals>Definite Integration

For,

x^{2} \neq n π + 1, n \in N

(the set of natural numbers), the integral

\int x \sqrt{\frac{2 \sin (x^{2} - 1) - \sin 2 (x^{2} - 1)}{2 \sin (x^{2} - 1) + \sin 2 (x^{2} - 1)}} d x

, is equal to
(where

c

is a constant of integration).

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

The integral

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

, is equal to

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

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,

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

\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

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

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

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

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

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

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:

EASY

Mathematics>Integral Calculus>Integrals>Definite Integration

\int_{0}^{\frac{π}{4}} x . \sec^{2} x d x =

EASY

Mathematics>Integral Calculus>Integrals>Definite Integration

The integral

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

is equal to

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

\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>Integrals>Definite Integration

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

EASY

Mathematics>Integral Calculus>Integrals>Definite Integration

The value of

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

is equal to

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

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

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

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)

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

\int_{0}^{k} \frac{d x}{2 + 18 x^{2}} = \frac{π}{24}

, then the value of

k

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

f (\frac{x - 4}{x + 2}) = 2 x + 1, (x \in R - \{1, - 2\})

, then

\int f (x) d x

is equal to

Let d⁡ d⁡ x F⁡ x = e sin x x , x > 0 . If ∫ 1 4 3 x e sin x 3 d⁡ x = F⁡ k - F⁡ 1 , then one of the possible values of k is

Important Questions on Integrals

Learn and Prepare for any exam you want !

Let $\frac{d}{d x} F (x) = (\frac{e^{sin x}}{x}), x > 0$ . If $\int_{1}^{4} \frac{3}{x} e^{sin x^{3}} d x = F (k) - F (1)$ , then one of the possible values of k is