Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

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Integrate with respect to $x$ :

$\frac{x^{2}}{x^{2} + x + 1}$

Important Questions on Indefinite Integration

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Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The value of

α

for which

4 α \int_{- 1}^{2} e^{- α |x|} d x = 5

, is

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Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

Let

α \in (0, \frac{π}{2})

, be constant. If the integral

\int \frac{t a n x + t a n α}{t a n x - \tan α} d x = A (x) c o s 2 α + B (x) s i n 2 α + C

, where C is a constant of integration, then the functions

A (x)

and

B (x)

are respectively

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Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

If

\int \frac{\cos x d x}{\sin^{3} x {(1 + \sin^{6} x)}^{\frac{2}{3}}} = f (x) {(1 + \sin^{6} x)}^{\frac{1}{λ}} + c

, where

c

is a constant of integration, then

λ f (\frac{π}{3})

is equal to

EASY

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

\int \frac{1}{x^{2} - 25} dx =

EASY

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

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

is equal to: (where

C

is an arbitarary constant)

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Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

\int \frac{2 x - 1}{(x - 1) (x + 2) (x - 3)} d x = A \log | x - 1 | + B \log | x + 2 | + C \log | x - 3 | + K,

then

A, B, C

are respectively

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The integral

\int \frac{d x}{{(x + 4)}^{\frac{8}{7}} {(x - 3)}^{\frac{6}{7}}}

is equal to: (where

C

is a constant of integration)

EASY

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

\int e^{x} secx (1 + tanx) dx =

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Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The integral

\int c o s (lnx) d x

, is equal to

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Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The integral

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

is equal to

EASY

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

\int \frac{1}{x + \sqrt{x}} dx =

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

If

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

then

α + \frac{1}{β} =

EASY

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

\int \sec^{2} (5 x - 1) d x =

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Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

\int \frac{1}{\sqrt{x} + x \sqrt{x}} d x =

EASY

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

\int \frac{1}{x (logx)} dx =

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

If

\int e^{2 x} (\cos x + 7 \sin x) d x = e^{2 x} g (x) + c

, where

c

is a constant of integration, then

g (0) + g (\frac{π}{2})

is equal to______.

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The integral $\int \frac{(2 sinθ - 1) cosθ}{5 - \cos^{2} θ - 4 sinθ} d θ$ is equal to

(where $C$ is a constant of integration)

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

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).

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Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

If

\int \frac{1}{(x^{2} + 4) (x^{2} + 9)} d x = A \tan^{- 1} (\frac{x}{2}) + B \tan^{- 1} (\frac{x}{3}) + C,

then

A - B =

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Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The value of the integral

\int \frac{e^{x} (x^{2} + 1)}{{(x + 1)}^{2}} d x =