Evaluate ∫ x 2 x 2 + 6 x + 12 d x .

x − 3 log | x 2 + 6 x + 12 | + 6 3 tan − 1 x + 3 3 + C

x 3 − 3 log | x 2 − 6 x | + 6 3 tan − 1 x + 3 3 + C

x 2 2 − 3 log | x 2 + 6 x + 12 | + 6 3 tan − 1 ( x + 3 3 ) + C

Evaluate ∫ x 2 x 2 + 6 x + 12 d x .

x − 3 log | x 2 + 6 x + 12 | + 6 3 tan − 1 x + 3 3 + 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

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

HARD

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Evaluate $\int \frac{x^{2}}{x^{2} + 6 x + 12} d x .$

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Important Questions on Indefinite Integration

HARD

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

\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>Integration of Rational Functions

\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>Integration of Rational Functions

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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

\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>Indefinite Integration>Integration of Rational Functions

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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

\int \sqrt{\frac{x - 5}{x - 7}} d x = A \sqrt{x^{2} - 12 x + 35} + \log |x - 6 + \sqrt{x^{2} - 12 x + 35}| + C

then

A =

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

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>Integration of Rational Functions

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>Integration of Rational Functions

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>Integration of Rational Functions

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>Indefinite Integration>Integration of Rational Functions

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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

\int \frac{(\sin x + \cos x) (2 - \sin 2 x)}{\sin^{2} 2 x} d x =

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

The integral

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

is equal to

HARD

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

Let

P (x) = \int \frac{d x}{e^{x} + 8 e^{- x} + 4 e^{- 3 x}},

Q (x) = \int \frac{d x}{e^{3 x} + 8 e^{x} + 4 e^{- x}}

and

R (x) = P (x) - 2 Q (x)

. If

R (x) = \frac{1}{2} A (\frac{B + 2 e^{- x}}{C}) + K,

then the value of

(A, B, C)

HARD

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

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>Integration of Rational Functions

The integral

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

is equal to

HARD

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

The integral

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

, is equal to

EASY

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

\int \frac{d x}{x^{2} + 4 x + 5} =

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Integration of Rational Functions

\int \frac{d x}{{(x^{2} - 2 x + 10)}^{2}} = A (\tan^{- 1} (\frac{x - 1}{3}) + \frac{f (x)}{x^{2} - 2 x + 10}) + C

, then (where

C

is a constant of integration)

Evaluate ∫ x 2 x 2 +6x+12 dx.

Important Questions on Indefinite Integration

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Evaluate $\int \frac{x^{2}}{x^{2} + 6 x + 12} d x .$