Indefinite Integrals

If \(\int {\left( {{e^{2x}} + 2{e^x} - {e^{ - x}} - 1} \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}}dx}\) = \(g\left( x \right){e^{\left( {{e^x} + {e^{ - x}}} \right)}} + c\)
where c is a constant of integration,
then g(0) is equal to :

If
\(\int {{{\cos \theta } \over {5 + 7\sin \theta - 2{{\cos }^2}\theta }}} d\theta\) = A\({\log _e}\left| {B\left( \theta \right)} \right| + C\),
where C is a constant of integration, then \({{{B\left( \theta \right)} \over A}}\)
can b...

The integral \(\int {{{\left( {{x \over {x\sin x + \cos x}}} \right)}^2}dx}\) is equal to
(where C is a constant of integration):

Let \(f\left( x \right) = \int {{{\sqrt x } \over {{{\left( {1 + x} \right)}^2}}}dx\left( {x \ge 0} \right)}\). Then f(3) – f(1) is eqaul to :

If \(\int {{{\sin }^{ - 1}}\left( {\sqrt {{x \over {1 + x}}} } \right)} dx\) = A(x)\({\tan ^{ - 1}}\left( {\sqrt x } \right)\) + B(x) + C,
where C is a constant of integration, then the
ordered pair (A(x), B(x)) can be :

For x2 \(\ne\) n\(\pi\) + 1, n \(\in\) N (the set of natural numbers), the integral \(\int {x\sqrt {{{2\sin ({x^2} - 1) - \sin 2({x^2} - 1)} \over {2\sin ({x^2} - 1) + \sin 2({x^2} - 1)}}} dx}\) is equal to :
(where c is a constant o...

If   \(f\left( x \right) = \int {{{5{x^8} + 7{x^6}} \over {{{\left( {{x^2} + 1 + 2{x^7}} \right)}^2}}}} \,dx,\,\left( {x \ge 0} \right),\)
\(f\left( 0 \right) = 0,\)    then the value of \(f(1)\) is :

The integral \(\int {{\rm{se}}{{\rm{c}}^{{\rm{2/ 3}}}}\,{\rm{x }}\,{\rm{cose}}{{\rm{c}}^{{\rm{4 / 3}}}}{\rm{x \,dx}}}\) is equal to
(Hence C is a constant of integration)

\(\int {{e^{\sec x}}}\) \((\sec x\tan xf(x) + \sec x\tan x + se{x^2}x)dx\)
= esecxf(x) + C then a possible choice of f(x) is :-

\(\int {{{\sin {{5x} \over 2}} \over {\sin {x \over 2}}}dx}\) is equal to
(where c is a constant of integration)

If \(\int {{{dx} \over {{x^3}{{(1 + {x^6})}^{2/3}}}} = xf(x){{(1 + {x^6})}^{{1 \over 3}}} + C}\)
where C is a constant of integration, then the
function ƒ(x) is equal to

The integral \(\int \,\)cos(loge x) dx is equal to : (where C is a constant of integration)

The integral \(\int {{{3{x^{13}} + 2{x^{11}}} \over {{{\left( {2{x^4} + 3{x^2} + 1} \right)}^4}}}} \,dx\) is equal to : (where C is a constant of integration)

The integral \(\int {{{2{x^3} - 1} \over {{x^4} + x}}} dx\) is equal to :
(Here C is a constant of integration)

Let \(a \in \left( {0,{\pi \over 2}} \right)\) be fixed. If the integral
\(\int {{{\tan x + \tan \alpha } \over {\tan x - \tan \alpha }}} dx\) = A(x) cos 2\(\alpha\) + B(x) sin 2\(\alpha\) + C, where C is a
constant of integration, then...

If  \(\int {{{\sqrt {1 - {x^2}} } \over {{x^4}}}}\) dx = A(x)\({\left( {\sqrt {1 - {x^2}} } \right)^m}\) + C, for a suitable chosen integer m and a function A(x), where C is a constant of integration, then (A(x))m equals :

If   \(\int {{{x + 1} \over {\sqrt {2x - 1} }}} \,dx\) = f(x) \(\sqrt {2x - 1}\) + C, where C is a constant of integration, then f(x) is equal to :

Let n \(\ge\) 2 be a natural number and \(0 < \theta < {\pi \over 2}.\) Then \(\int {{{{{\left( {{{\sin }^n}\theta - \sin \theta } \right)}^{1/n}}\cos \theta } \over {{{\sin }^{n + 1}}\theta }}} \,d\theta\) is equal to - (where C is a...

If  \(\int \,\)x5.e\(-\)4x3 dx = \({1 \over {48}}\)e\(-\)4x3 f(x) + C, where C is a constant of inegration, then f(x) is equal to -

If \(\int {{{dx} \over {{{\left( {{x^2} - 2x + 10} \right)}^2}}}} = A\left( {{{\tan }^{ - 1}}\left( {{{x - 1} \over 3}} \right) + {{f\left( x \right)} \over {{x^2} - 2x + 10}}} \right) + C\)
where C is a constant of integration then :

If \(\int {{x^5}} {e^{ - {x^2}}}dx = g\left( x \right){e^{ - {x^2}}} + c\), where c is a constant of integration, then \(g\)(–1) is equal to :

If \(\int {{{\tan x} \over {1 + \tan x + {{\tan }^2}x}}dx = x - {K \over {\sqrt A }}{{\tan }^{ - 1}}}\) \(\left( {{{K\,\tan x + 1} \over {\sqrt A }}} \right) + C,(C\,\,\) is a constant of integration) then the ordered pair (K, A) is equal ...

If \(f\left( {{{x - 4} \over {x + 2}}} \right) = 2x + 1,\) (x \(\in\) R \(-\){1, \(-\) 2}), then \(\int f \left( x \right)dx\) is equal to :
(where C is a constant of integration)

If    \(\int {{{2x + 5} \over {\sqrt {7 - 6x - {x^2}} }}} \,\,dx = A\sqrt {7 - 6x - {x^2}} + B{\sin ^{ - 1}}\left( {{{x + 3} \over 4}} \right) + C\)
(where C is a constant of integration), then the ordered pair (A, B) is equal to :

The integral
\(\int {{{{{\sin }^2}x{{\cos }^2}x} \over {{{\left( {{{\sin }^5}x + {{\cos }^3}x{{\sin }^2}x + {{\sin }^3}x{{\cos }^2}x + {{\cos }^5}x} \right)}^2}}}} dx\)
is equal to

If \(\,\,\,\) f\(\left( {{{3x - 4} \over {3x + 4}}} \right)\) = x + 2, x \(\ne\) \(-\) \({4 \over 3}\), and
\(\int {}\)f(x) dx = A log\(\left| {} \right.\)1 \(-\) x \(\left| {} \right.\) + Bx + C,
then the ordered pair (A, B) is eq...

The integral
\(\int {\sqrt {1 + 2\cot x(\cos ecx + \cot x)\,} \,\,dx}\)
\(\left( {0 < x < {\pi \over 2}} \right)\) is equal to :
(where C is a constant of integration)

Let \({I_n} = \int {{{\tan }^n}x\,dx} ,\,\left( {n > 1} \right).\)
If \({I_4} + {I_6}\) = \(a{\tan ^5}x + b{x^5} + C\), where C is a constant of integration,
then the ordered pair \(\left( {a,b} \right)\) is equal to

If   \(\int {{{dx} \over {{{\cos }^3}x\sqrt {2\sin 2x} }}} = {\left( {\tan x} \right)^A} + C{\left( {\tan x} \right)^B} + k,\)
where k is a constant of integration, then A + B +C equals :

The integral \(\int {{{dx} \over {\left( {1 + \sqrt x } \right)\sqrt {x - {x^2}} }}}\) is equal to :
(where C is a constant of integration.)

The integral \(\int {{{2{x^{12}} + 5{x^9}} \over {{{\left( {{x^5} + {x^3} + 1} \right)}^3}}}} dx\) is equal to :

The integral \(\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}}\) equals :

The integral \(\int {\left( {1 + x - {1 \over x}} \right){e^{x + {1 \over x}}}dx}\) is equal to

If \(\int {f\left( x \right)dx = \psi \left( x \right),}\) then \(\int {{x^5}f\left( {{x^3}} \right)dx}\) is equal to

If the \(\int {{{5\tan x} \over {\tan x - 2}}dx = x + a\,\ln \,\left| {\sin x - 2\cos x} \right| + k,}\) then \(a\) is
equal to :

The value of \(\sqrt 2 \int {{{\sin xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}}\) is

\(\int {{{dx} \over {\cos x + \sqrt 3 \sin x}}}\) equals

\(\int {{{\left\{ {{{\left( {\log x - 1} \right)} \over {1 + {{\left( {\log x} \right)}^2}}}} \right\}}^2}\,\,dx}\) is equal to

\(\int {{{dx} \over {\cos x - \sin x}}}\) is equal to

If \(\int {{{\sin x} \over {\sin \left( {x - \alpha } \right)}}dx = Ax + B\log \sin \left( {x - \alpha } \right), + C,}\) then value of
\((A, B)\) is