Indefinite Integrals

Let $f(x) = \int \left( \frac{16x + 24}{x^2 + 2x - 15} \right) dx$. If $f(4) = 14 \log_e(3)$ and $f(7) = \log_e(2^\alpha \cdot 3^\beta)$, $\alpha, \beta \in \mathbb{N}$, then $\alpha + \beta$ is equal to :

If $\int\left(\frac{1-5 \cos ^2 x}{\sin ^5 x \cos ^2 x}\right) d x=f(x)+\mathrm{C}$, where C is the constant of integration, then $f\left(\frac{\pi}{6}\right)-f\left(\frac{\pi}{4}\right)$ is equal to

Let \(f(x)=\int \frac{d x}{x^{2 / 3}+2 x^{1 / 2}},\) be such that $f(0) = -26 + 24 \log_e(2)$. If $f(1) = a + b \log_e(3)$, where $a, b \in \mathbb{Z}$, then $a + b$ is equal to :

Let $f(t)=\int\left(\frac{1-\sin \left(\log _e t\right)}{1-\cos \left(\log _e t\right)}\right) d t, t>1$.
If $f\left(e^{\pi / 2}\right)=-e^{\pi / 2}$ and $f\left(e^{\pi / 4}\right)=\alpha e^{\pi / 4}$, then $\alpha$ equals

Let $f(x)=\int \frac{\left(2-x^2\right) \cdot \mathrm{e}^x}{(\sqrt{1+x})(1-x)^{3 / 2}} \mathrm{~d} x$. If $f(0)=0$, then $f\left(\frac{1}{2}\right)$ is equal to:

Let $\mathrm{I}(x)=\int \frac{3 d x}{(4 x+6)\left(\sqrt{4 x^2+8 x+3}\right)}$ and $\mathrm{I}(0)=\frac{\sqrt{3}}{4}+20$. If
$\mathrm{I}\left(\frac{1}{2}\right)=\frac{a \sqrt{2}}{b}+\mathrm{c}$, where $a, b, \mathrm{c} \in \mathrm{N}, \oper...

If $\int(\sin x)^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}} d x= -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}}-\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}}-\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}}+\frac{p_4}{q_4}(\cot x)^{\frac{-3}{2}}+\mathrm{C}$, where $p_i$ ...

If $\int\left(\frac{1}{x}+\frac{1}{x^3}\right)\left(\sqrt[23]{3 x^{-24}+x^{-26}}\right) \mathrm{d} x=-\frac{\alpha}{3(\alpha+1)}\left(3 x^\beta+x^\gamma\right)^{\frac{\alpha+1}{\alpha}}+C, x>0,(\alpha, \beta, \gamma \in \mathbf{Z})$, where ...

If $\int \frac{\left(\sqrt{1+x^2}+x\right)^{10}}{\left(\sqrt{1+x^2}-x\right)^9} \mathrm{~d} x=\frac{1}{\mathrm{~m}}\left(\left(\sqrt{1+x^2}+x\right)^{\mathrm{n}}\left(\mathrm{n} \sqrt{1+x^2}-x\right)\right)+\mathrm{C}$ where C is the consta...

\(\text { Let } f(x)=\int x^3 \sqrt{3-x^2} d x \text {. If } 5 f(\sqrt{2})=-4 \text {, then } f(1) \text { is equal to }\)

If $f(x)=\int \frac{1}{x^{1 / 4}\left(1+x^{1 / 4}\right)} \mathrm{d} x, f(0)=-6$, then $f(1)$ is equal to :

If $\int \frac{2 x^2+5 x+9}{\sqrt{x^2+x+1}} \mathrm{~d} x=x \sqrt{x^2+x+1}+\alpha \sqrt{x^2+x+1}+\beta \log _{\mathrm{e}}\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|+\mathrm{C}$, where $C$ is the constant of integration, then $\alpha+2 \beta$ ...

Let $\mathrm{I}(x)=\int \frac{d x}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $\mathrm{I}(37)-\mathrm{I}(24)=\frac{1}{4}\left(\frac{1}{\mathrm{~b}^{\frac{1}{13}}}-\frac{1}{\mathrm{c}^{\frac{1}{13}}}\right), \mathrm{b}, \mathrm{c} \i...

Let $\int x^3 \sin x \mathrm{~d} x=g(x)+C$, where $C$ is the constant of integration. If $8\left(g\left(\frac{\pi}{2}\right)+g^{\prime}\left(\frac{\pi}{2}\right)\right)=\alpha \pi^3+\beta \pi^2+\gamma, \alpha, \beta, \gamma \in Z$, then $\a...

If $\int \mathrm{e}^x\left(\frac{x \sin ^{-1} x}{\sqrt{1-x^2}}+\frac{\sin ^{-1} x}{\left(1-x^2\right)^{3 / 2}}+\frac{x}{1-x^2}\right) \mathrm{d} x=\mathrm{g}(x)+\mathrm{C}$, where C is the constant of integration, then $g\left(\frac{1}{2}\r...

Let \(\int \frac{2-\tan x}{3+\tan x} \mathrm{~d} x=\frac{1}{2}\left(\alpha x+\log _e|\beta \sin x+\gamma \cos x|\right)+C\), where \(C\) is the constant of integration. Then \(\alpha+\frac{\gamma}{\beta}\) is equal to :

Let \(I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x\). If \(I(0)=3\), then \(I\left(\frac{\pi}{12}\right)\) is equal to

If \(\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}} \mathrm{~d} x=\mathrm{A}\left(\frac{\alpha x-1}{\beta x+3}\right)^B+\mathrm{C}\), where \(\mathrm{C}\) is the constant of integration, then the value of \(\alpha+\beta+20 \mathrm{AB}\) is _______...

If \(\int \frac{1}{\mathrm{a}^2 \sin ^2 x+\mathrm{b}^2 \cos ^2 x} \mathrm{~d} x=\frac{1}{12} \tan ^{-1}(3 \tan x)+\) constant, then the maximum value of \(\mathrm{a} \sin x+\mathrm{b} \cos x\), is :

If \(\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cosec}^2 x+\frac{3}{2}\right)+\beta \log _x\left|\tan \frac{x}{2}\right|+\mathrm{C}\)
where \(\alpha, \beta \in \mathbb{R}\) and \(\mathrm{C}\) i...

For \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), if \(y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x\), and \(\lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0\) th...

If \(\int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x=A \sqrt{\cos \theta \tan x-\sin \theta}+B \sqrt{\cos \theta-\sin \theta \cot x}+C\), where \(C\) is the integration constant, then ...

\(\text { The integral } \int \frac{\left(x^8-x^2\right) \mathrm{d} x}{\left(x^{12}+3 x^6+1\right) \tan ^{-1}\left(x^3+\frac{1}{x^3}\right)} \text { is equal to : }\)

Let \(I(x)=\int \frac{(x+1)}{x\left(1+x e^{x}\right)^{2}} d x, x > 0\). If \(\lim_\limits{x \rightarrow \infty} I(x)=0\), then \(I(1)\) is equal to :

The integral \(\int\left[\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right] \ln \left(\frac{e x}{2}\right) d x\) is equal to :

Let \(I(x)=\int \frac{x^{2}\left(x \sec ^{2} x+\tan x\right)}{(x \tan x+1)^{2}} d x\). If \(I(0)=0\), then \(I\left(\frac{\pi}{4}\right)\) is equal to :

If $\int \sqrt{\sec 2 x-1} d x=\alpha \log _e\left|\cos 2 x+\beta+\sqrt{\cos 2 x\left(1+\cos \frac{1}{\beta} x\right)}\right|+$ constant, then $\beta-\alpha$ is equal to ____________.

Let \(f(x) = \int {{{2x} \over {({x^2} + 1)({x^2} + 3)}}dx}\). If \(f(3) = {1 \over 2}({\log _e}5 - {\log _e}6)\), then \(f(4)\) is equal to

Let $f(x)=\int \frac{d x}{\left(3+4 x^{2}\right) \sqrt{4-3 x^{2}}},|x|<\frac{2}{\sqrt{3}}$. If $f(0)=0$ and $f(1)=\frac{1}{\alpha \beta} \tan ^{-1}\left(\frac{\alpha}{\beta}\right)$,

$\alpha, \beta>0$, then $\alpha^{2}+\beta^{2}$ is equal ...

Let \(I(x)=\int \sqrt{\frac{x+7}{x}} \mathrm{~d} x\) and \(I(9)=12+7 \log _{e} 7\). If \(I(1)=\alpha+7 \log _{e}(1+2 \sqrt{2})\), then \(\alpha^{4}\) is equal to _________.

If \(I(x) = \int {{e^{{{\sin }^2}x}}(\cos x\sin 2x - \sin x)dx}\) and \(I(0) = 1\), then \(I\left( {{\pi \over 3}} \right)\) is equal to :

For \(\alpha, \beta, \gamma, \delta \in \mathbb{N}\), if $$\int\left(\left(\frac{x}{e}\right)^{2 x}+\left(\frac{e}{x}\right)^{2 x}\right) \log _{e} x d x=\frac{1}{\alpha}\left(\frac{x}{e}\right)^{\beta x}-\frac{1}{\gamma}\left(\frac{e}{x}\r...

For \(I(x)=\int \frac{\sec ^{2} x-2022}{\sin ^{2022} x} d x\), if \(I\left(\frac{\pi}{4}\right)=2^{1011}\), then

If \(\int {{{({x^2} + 1){e^x}} \over {{{(x + 1)}^2}}}dx = f(x){e^x} + C}\), where C is a constant, then \({{{d^3}f} \over {d{x^3}}}\) at x = 1 is equal to :

If \(\int {{1 \over x}\sqrt {{{1 - x} \over {1 + x}}} dx = g(x) + c}\), \(g(1) = 0\), then \(g\left( {{1 \over 2}} \right)\) is equal to :

\(\text { The integral } \int \frac{\left(1-\frac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt{3}} \sin 2 x\right)} d x \text { is equal to }\)

The integral \(\int {{1 \over {\root 4 \of {{{(x - 1)}^3}{{(x + 2)}^5}} }}} \,dx\) is equal to : (where C is a constant of integration)

If \(\int {{{\sin x} \over {{{\sin }^3}x + {{\cos }^3}x}}dx = }\)
\(\alpha {\log _e}|1 + \tan x| + \beta {\log _e}|1 - \tan x + {\tan ^2}x| + \gamma {\tan ^{ - 1}}\left( {{{2\tan x - 1} \over {\sqrt 3 }}} \right) + C\), when C is constant...

If \(\int {{{dx} \over {{{({x^2} + x + 1)}^2}}} = a{{\tan }^{ - 1}}\left( {{{2x + 1} \over {\sqrt 3 }}} \right) + b\left( {{{2x + 1} \over {{x^2} + x + 1}}} \right) + C}\), x > 0 where C is the constant of integration, then the value of $$...

If \(\int {{{2{e^x} + 3{e^{ - x}}} \over {4{e^x} + 7{e^{ - x}}}}dx = {1 \over {14}}(ux + v{{\log }_e}(4{e^x} + 7{e^{ - x}})) + C}\), where C is a constant of integration, then u + v is equal to _____________.

The value of the integral \(\int {{{\sin \theta .\sin 2\theta ({{\sin }^6}\theta + {{\sin }^4}\theta + {{\sin }^2}\theta )\sqrt {2{{\sin }^4}\theta + 3{{\sin }^2}\theta + 6} } \over {1 - \cos 2\theta }}} \,d\theta\) is :

The integral \(\int {{{{e^{3{{\log }_e}2x}} + 5{e^{2{{\log }_e}2x}}} \over {{e^{4{{\log }_e}x}} + 5{e^{3{{\log }_e}x}} - 7{e^{2{{\log }_e}x}}}}} dx\), x > 0, is equal to : (where c is a constant of integration)

If \(\int {{{\cos x - \sin x} \over {\sqrt {8 - \sin 2x} }}} dx = a{\sin ^{ - 1}}\left( {{{\sin x + \cos x} \over b}} \right) + c\), where c is a constant of integration, then
the ordered pair (a, b) is equal to :

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

The integral \(\int {{{(2x - 1)\cos \sqrt {{{(2x - 1)}^2} + 5} } \over {\sqrt {4{x^2} - 4x + 6} }}} dx\) is equal to (where c is a constant of integration)

For real numbers \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\), if \(\int {{{({x^2} - 1) + {{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)} \over {({x^4} + 3{x^2} + 1){{\tan }^{ - 1}}\left( {{{{x^2} + 1} \over x}} \right)}}dx}\)...

The integral \(\int {{{dx} \over {{{(x + 4)}^{{8 \over 7}}}{{(x - 3)}^{{6 \over 7}}}}}}\) is equal to :
(where C is a constant of integration)

If ƒ'(x) = tan–1(secx + tanx), \(- {\pi \over 2} < x < {\pi \over 2}\),
and
ƒ(0) = 0, then ƒ(1) is equal to :

If \(\int {{{d\theta } \over {{{\cos }^2}\theta \left( {\tan 2\theta + \sec 2\theta } \right)}}} = \lambda \tan \theta + 2{\log _e}\left| {f\left( \theta \right)} \right| + C\)
where C is a constant of integration, then the
ordered pair...

If \(\int {{{\cos xdx} \over {{{\sin }^3}x{{\left( {1 + {{\sin }^6}x} \right)}^{2/3}}}}} = f\left( x \right){\left( {1 + {{\sin }^6}x} \right)^{1/\lambda }} + c\)
where c is a constant of integration, then $$\lambda f\left( {{\pi \over 3}...