Indefinite Integrals

The integral $\int \frac{\sec ^2 x}{(\sec x+\tan x)^{9 / 2}} d x$ equals (for some arbitrary constant \(K\))

Let \(I = \int {{{{e^x}} \over {{e^{4x}} + {e^{2x}} + 1}}dx,\,\,J = \int {{{{e^{ - x}}} \over {{e^{ - 4x}} + {e^{ - 2x}} + 1}}dx.} }\) Then
for an arbitrary constant \(C\), the value of \(J -I\) equals :

Let \(f(x)=\frac{x}{\left(1+x^{n}\right)^{1 / n}}\) for \(n \geq 2\) and \(g(x)=\underbrace{(f o f o \ldots . o f)}_{f \text { occurs } n \text { times }}(x)\). Then \(\int x^{n-2} g(x) d x\) equals :

Let \(F(x)\) be an indefinite integral of \(si{n^2}x.\)
STATEMENT-1: The function \(F(x)\) satisfies \(F\left( {x + \pi } \right) = F\left( x \right)\)
for all real \(x\). because
STATEMENT-2: $${\sin ^2}\left( {x + \pi } \right) = {\sin ...

$\int \frac{x^2-1}{x^3 \sqrt{2 x^4-2 x^2+1}} d x$ is equal to

If \(\int\limits_{\sin x}^1 {{t^2}f\left( t \right)dt = 1 - \sin x,}\) then f\(\left( {{1 \over {\sqrt 3 }}} \right)\) is

For any natural number \(m\), evaluate
\(\int {\left( {{x^{3m}} + {x^{2m}} + {x^m}} \right){{\left( {2{x^{2m}} + 3{x^m} + 6} \right)}^{l/m}}dx,x > 0.}\)

Evaluate \(\int {{{\sin }^{ - 1}}\left( {{{2x + 2} \over {\sqrt {4{x^2} + 8x + 13} }}} \right)} \,dx.\)

Integrate \(\int {{{{x^3} + 3x + 2} \over {{{\left( {{x^2} + 1} \right)}^2}\left( {x + 1} \right)}}dx.}\)

Evaluate \(\int {{{\left( {x + 1} \right)} \over {x{{\left( {1 + x{e^x}} \right)}^2}}}dx}\).

The value of the integral \(\int {{{{{\cos }^3}x + {{\cos }^5}x} \over {{{\sin }^2}x + {{\sin }^4}x}}} \,dx\,\) is

Find the indefinite integral \(\,\int {\cos 2\theta {\mkern 1mu} ln\left( {{{\cos \theta + \sin \theta } \over {\cos \theta - \sin \theta }}} \right)} {\mkern 1mu} d\theta\)

Find the indefinite integral \(\int {\left( {{1 \over {\root 3 \of x + \root 4 \of 4 }} + {{In\left( {1 + \root 6 \of x } \right)} \over {\root 3 \of x + \root \, \of x }}} \right)} dx\)

If \(\int {{{4{e^x} + 6{e^{ - x}}} \over {9{e^x} - 4{e^{ - x}}}}\,dx = Ax + B\,\,\log \left( {9{e^{2x}} - 4} \right) + C,}\) then
\(A = .....,B = .....\) and \(C = .....\)

Evaluate \(\int {\left( {\sqrt {\tan x} + \sqrt {\cot x} } \right)dx}\)

Evaluate :\(\,\,\int {\left[ {{{{{\left( {\cos 2x} \right)}^{1/2}}} \over {\sin x}}} \right]dx}\)

Evaluate the following \(\int {\sqrt {{{1 - \sqrt x } \over {1 + \sqrt x }}dx} }\)

Evaluate the following \(\int {{{dx} \over {{x^2}{{\left( {{x^4} + 1} \right)}^{3/4}}}}}\)

Evaluate : \(\int {{{\left( {x - 1} \right){e^x}} \over {{{\left( {x + 1} \right)}^3}}}dx}\)

Evaluate \(\int {\left( {{e^{\log x}} + \sin x} \right)\cos x\,\,dx.}\)

Evaluate \(\int {{{{x^2}dx} \over {{{\left( {a + bx} \right)}^2}}}}\)

Evaluate \(\int {{{\sin x} \over {\sin x - \cos x}}dx}\)