If m is a non-zero number and ∫ x 5 m - 1 + 2 x 4 m - 1 x 2 m + x m + 1 3 d x = f x + c , then f x is equal to

x 5 m - x 4 m 2 m x 2 m + x m + 1 2

2 m x 5 m + x 4 m x 2 m + x m + 1 2

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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If $\int \frac{sin x}{sin (x - π / 4)} dx = A (f (x) + log |sin x - cos x|) + C$ then

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Important Questions on Indefinite Integration

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Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

\int \frac{1}{\sqrt{9 - 16 x^{2}}} d x = α \sin^{- 1} (β x) + c,

then

α + \frac{1}{β} =

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

\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>Indefinite Integration>Methods of Integration by Substitution

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>Indefinite Integration>Methods of Integration by Substitution

The integral

\int \sqrt{1 + 2 \cot x (cosec x + \cot x)} d x, (0 < x < \frac{π}{2})

is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

\int \frac{d x}{\sqrt{16 - 9 x^{2}}} = A \sin^{- 1} (B x) + C

then

A + B =

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

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

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

The integral

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

, is equal to

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

The integral

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

is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

\int \frac{1}{\sin x . \cos^{2} x} d x =

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

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>Indefinite Integration>Methods of Integration by Substitution

m

is a non-zero number and

\int \frac{x^{5 m - 1} + 2 x^{4 m - 1}}{{(x^{2 m} + x^{m} + 1)}^{3}} d x = f (x) + c,

then

f (x)

is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

The integral

\int \frac{\sin^{2} x \cos^{2} x}{{(\sin^{5} x + \cos^{3} x \sin^{2} x + \sin^{3} x \cos^{2} x + \cos^{5} x)}^{2}} d x

, is equal to

(where

C

is the constant of integration).

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

\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>Indefinite Integration>Methods of Integration by Substitution

The integral

\int \frac{2 x^{12} + 5 x^{9}}{{(x^{5} + x^{3} + 1)}^{3}} d x

, is equal to

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

\int \frac{d x}{x^{3} {(1 + x^{6})}^{\frac{2}{3}}} = x f (x) {(1 + x^{6})}^{\frac{1}{3}} + C

, where

C

is a constant of integration, then the function

f (x)

is equal to

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

\int \frac{d x}{\cos^{3} x \sqrt{2 \sin 2 x}} = {(\tan x)}^{A} + C {(\tan x)}^{B} + k

, where

k

is a constant of integration, then

A + B + C

equals

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

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>Indefinite Integration>Methods of Integration by Substitution

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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration by Substitution

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).

If ∫ sin x sin x - π / 4 dx = A f x + log sin x - cos x + C then

Important Questions on Indefinite Integration

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If $\int \frac{sin x}{sin (x - π / 4)} dx = A (f (x) + log |sin x - cos x|) + C$ then