Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

EASY

Earn 100

Show that $F (x)$ is an antiderivative of $f (x)$ .

$f (x) = 2 x + 1$ and $F (x) = x^{2} + x - 3.2$

Important Questions on Indefinite Integration

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

If

\int f (x) \sin x \cos x d x = \frac{1}{2 (b^{2} - a^{2})} \log f (x) + c

where

c

is the constant of integration, then

f (x)

is equal to

HARD

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

\int \frac{s i n \frac{5 x}{2}}{s i n \frac{x}{2}} d x

, is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

\int \frac{3^{x}}{\sqrt{1 - 9^{x}}} d x =

EASY

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

$\int_{0}^{π / 2} (x - [\sin x]) d x =$ ________.

(where $[x] =$ greatest integer not greater than $x$ ).

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

Let

x = f^{''} (t) \cos t + f^{'} (t) \sin t

and

y = - f^{''} (t) \sin t + f^{'} (t) \cos t

then

\int {[{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}]}^{1 / 2} d t

is:

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

\int \frac{\sin^{2} x - \cos^{2} x}{\sin^{2} x \cos^{2} x} d x

is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

If

f^{'} (x) = a \sin x + b \cos x, f^{'} (0) = 4, f (0) = 3

and

f (\frac{π}{2}) = 5,

then

f (x) =

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

For '

c

' is an arbitrary constant,

\int \frac{1}{\sqrt{(1 + x)} (1 + x)^{\frac{1}{3}}} d x =

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

If

f (\frac{3 x - 4}{3 x + 4}) = x + 2, x \neq - \frac{4}{3}

, and

\int f (x) d x = A \log |1 - x| + B x + C

, then the ordered pair

(A, B)

is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

\int \frac{\sin^{8} x - \cos^{8} x}{(1 - 2 \sin^{2} x \cos^{2} x)} d x

is equal to

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

If

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

,

then

f (\frac{π}{2})

is (where

C

is an arbitrary constant)

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

If

f^{'} (x) = 2 - \frac{5}{x^{4}}

and

f (1) = \frac{14}{3},

then

f (- 1) =

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

For '

c

' is an arbitrary constant,

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

HARD

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

Let

S

be the set of real numbers

p

such that there is no non-zero continuous function

f : R \to R

satisfying

\int_{0}^{x} f (t) d t = p f (x)

for all

x \in R

. Then,

S

is

EASY

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

For '

c

' is an arbitrary constant,

\int e^{\log x} d x =

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

Evaluate:

\int \frac{\cos 2 x + 2 \sin^{2} x}{\cos^{2} x} d x

.

EASY

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

\int_{0}^{2} x e^{x} d x =

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

If

\int e^{\sec x} (\sec x \tan x f (x) + (\sec x \tan x + \sec^{2} x)) d x = e^{\sec x} f (x) + C,

then a possible choice of

f (x)

is:

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

Suppose that

F (x)

is an anti-derivative of

f (x) = \frac{\sin x}{x},

where

x > 0,

then

\int_{1}^{3} \frac{\sin 2 x}{x} d x

can be expressed as:

MEDIUM

Mathematics>Integral Calculus>Indefinite Integration>Basics of Indefinite Integral

If

\int \frac{2 x^{2}}{(2 x^{2} + α) (x^{2} + 5)} d x = \frac{\sqrt{5}}{3} \tan^{- 1} \frac{x}{\sqrt{5}} - \frac{\sqrt{2}}{3} \tan^{- 1} \frac{x}{\sqrt{2}} + c

, then

α =