Integrate 1 + x 2 / 3 1 + x with respect to x.

3 2 x 2 / 3 + 2 log x 1 / 3 + 1 + 1 2 log x 2 / 3 - x 1 / 3 + 1 - 3 tan -1 2 x 1 / 3 - 1 3 + c

3 2 x 2 / 3 - 4 log x 1 / 3 - 1 + 1 2 log x 2 / 3 - x 1 / 3 - 1 + 3 tan -1 4 x 1 / 3 - 1 3 - c

4 2 x 2 / 3 - 2 log x 1 / 3 - 3 + 1 2 log x 2 / 3 - x 1 / 3 - 2 + 3 tan -1 4 x 1 / 3 - 1 3 - c

4 2 x 2 / 3 - 2 log x 1 / 3 - 3 - 1 2 log x 2 / 3 + x 1 / 3 - 4 - 3 tan -1 2 x 1 / 3 + 1 3 - c

Integrate 1 + x 2 / 3 1 + x with respect to x.

3 2 x 2 / 3 + 2 log x 1 / 3 + 1 + 1 2 log x 2 / 3 - x 1 / 3 + 1 - 3 tan -1 2 x 1 / 3 - 1 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

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

HARD

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Integrate $\frac{1 + x^{2 / 3}}{1 + x}$ with respect to x.

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Important Questions on Indefinite Integration

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of 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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of 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

HARD

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 =

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of 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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of 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

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

\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 =

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

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

The integral

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

, is equal to

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

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

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>Methods of Integration

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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of 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

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)

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

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

then

α + \frac{1}{β} =

MEDIUM

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

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).

HARD

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

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

\int f (x) d x = ψ (x)

, then

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

, is equal to

EASY

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

The integral

\int c o s (lnx) d x

, is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Methods of Integration

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

Integrate 1 + x 2 / 3 1 + x with respect to x.

Important Questions on Indefinite Integration

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Integrate $\frac{1 + x^{2 / 3}}{1 + x}$ with respect to x.