Indefinite Integration

\(\int \frac{\mathrm{e}^{2030 \log x}-\mathrm{e}^{2029 \log x}}{\mathrm{e}^{2028 \log x}-\mathrm{e}^{2027 \log x}} \mathrm{~d} x=\ldots\)

\(\int \frac{\mathrm{d} x}{2 \mathrm{e}^{2 x}+3 \mathrm{e}^x+1}=\)

\(\int \frac{x+\sin x}{1+\cos x} d x=\)

\(\int \frac{x \mathrm{~d} x}{(x-1)(x-2)}=\)

If $\int \tan ^4 x \mathrm{~d} x=\mathrm{a} \tan ^3 x+\mathrm{b} \tan x+\mathrm{c} x+\mathrm{k}$ (where k is the constant of integration) then the value of $\mathrm{a}-\mathrm{b}+\mathrm{c}=$

The value of $I=\int \frac{(x-1) \mathrm{e}^x}{(x+1)^3} \mathrm{dx}$ is

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

If $\int \frac{d x}{\sqrt[3]{\sin ^{11} x \cos x}}=-\left(\frac{3}{8} f(x)+\frac{3}{2} g(x)\right)+c$ then

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

The value of $\int \frac{x+1}{x\left(1+x \mathrm{e}^x\right)^2} \mathrm{dx}$ is equal to

The value of $\int \frac{\mathrm{d} x}{(x+1)^{3 / 4}(x-2)^{5 / 4}}$ is equal to

\(\int \sqrt{\mathrm{e}^x-1} \mathrm{dx}=\)

If $\int \frac{\cos x-\sin x}{\sqrt{8-\sin 2 x}} d x=a \sin ^{-1}\left(\frac{\sin x+\cos x}{b}\right)+c$
Where c is a constant of integration, then the ordered pair $(\mathrm{a}, \mathrm{b})$ is equal to

The integral $\int \sec ^{\frac{2}{3}} x \cdot \operatorname{cosec}^{\frac{4}{3}} x \mathrm{~d} x$ is equal to

If $I=\int e^{\sin \theta}\left(\log \sin \theta+\operatorname{cosec}^2 \theta\right) \cos \theta d \theta$, then $I$ is equal to

The value of $\mathrm{I}=\int \frac{x^2}{(\mathrm{a}+\mathrm{bx})^2} \mathrm{dx}$ is

$\int\left(1+x-\frac{1}{x}\right) e^{x+\frac{1}{x}} d x$ equal to

\(\int \tan ^{-1}\left(\frac{1-\sin x}{1+\sin x}\right) d x=\)

\(\int \frac{\left(x^2+1\right)}{(x+1)^2} \mathrm{~d} x=\)

The value of $\int \frac{\mathrm{d} x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}$ is

If $\int\left(\frac{4 e^x-25}{2 e^x-5}\right) d x=A x+B \log \left(2 e^x-5\right)+c \quad$ (where c is a constant of integration) then

The value of $\int \frac{(x-1) \mathrm{e}^x}{(x+1)^3} \mathrm{~d} x$ is equal to

If $\quad \int(2 x+4) \sqrt{x-1} d x=a(x-1)^{5 / 2}+b(x-1)^{3 / 2}+c$ where $c$ is a constant of integration, then the value of $(2 a+b)$ is

\(\int \frac{2 x^2-1}{\left(x^2+4\right)\left(x^2-3\right)} d x=\)

If $\int \frac{\mathrm{d} x}{\cos ^3 x \sqrt{2 \sin 2 x}}=(\tan x)^A+C(\tan x)^B+K$, where K is a constant of integration, then the value of $5(A+B+C)$ is equal to

The value of $\int \mathrm{e}^x\left(\frac{1-\sin x}{1-\cos x}\right) \mathrm{dx}$ is equal to

\(\int \frac{\operatorname{cosec} x d x}{\cos ^2\left(1+\log \tan \frac{x}{2}\right)}=\)

The value of $\int \sin \sqrt{x} \mathrm{dx}$ is equal to

If $\mathrm{f}\left(\frac{x-4}{x-2}\right)=2 x+1, x \in \mathbb{R}-\{1,-2\}$, then $\int \mathrm{f}(x) \mathrm{d} x$ is equal to

\(\int\left(\frac{x+2}{x+4}\right)^2 \cdot e^x \mathrm{~d} x=\)

$\int \frac{\mathrm{d} x}{3-2 \cos 2 x}=\frac{\tan ^{-1}(\mathrm{f}(x))}{\sqrt{5}}+\mathrm{c}$, (where c is a constant of integration), then $f(\pi / 4)$ has the value

\(\int \log (1+x)^{1+x} \mathrm{~d} x=\)

\(\int 3^{3^x} \cdot 3^x d x=\)

$\int\left(\mathrm{f}(x) \mathrm{g}^{\prime \prime}(x)-\mathrm{f}^{\prime \prime}(x) \mathrm{g}(x)\right) \mathrm{d} x$ is equal to

$\int \frac{\log \sqrt{x}}{3 x} \mathrm{dx}$ is equal to

If $\int \frac{x+1}{\sqrt{2 x-1}} \mathrm{~d} x=\mathrm{f}(x) \sqrt{2 x-1}+\mathrm{c}$, (where c is a constant of integration), then $\mathrm{f}(x)$ is equal to

The value of $\mathrm{I}=\int \frac{\mathrm{d} x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}$ is

If $\int \frac{\mathrm{d} x}{\cos ^3 x \sqrt{2 \sin 2 x}}=(\tan x)^A+C(\tan x)^B+\mathrm{k}$
where k is a constant of integration, then $A+B+C$ equals

If $\int \frac{\log \left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}} d t=\frac{1}{2}(g(t))^2+c$
where c is a constant of integration, then $\mathrm{g}(2)$ is equal to

If $f(x)=\frac{\sin ^2 \pi x}{1+\pi^x}$, then $\int(f(x)+f(-x)) d x$ is equal to

The integral $\int \frac{2 x^3-1}{x^4+x} \mathrm{~d} x$ is equal to

\(\int \frac{x^3-7 x+6}{x^2+3 x} \mathrm{~d} x=\)

\(\int \frac{\mathrm{e}^{\tan ^{-1} x}}{1+x^2}\left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] \mathrm{d} x,\)
where $x>0$ is

\(\int \cos ^{\frac{-3}{7}} x \cdot \sin ^{\frac{-11}{7}} x d x=\)

$\int \frac{x^2-4}{x^4+9 x^2+16} \mathrm{dx}=\tan ^{-1}(\mathrm{f}(x))+\mathrm{c}$ (where c is a constant of integration), then value of $f(2)$ is

The value of $\int \frac{\sec x \cdot \tan x}{9-16 \tan ^2 x} \mathrm{dx}$ is equal to

The value of $\int \frac{\mathrm{d} x}{7+6 x-x^2}$ is equal to

The value of $\int \frac{d x}{5+4 \sin x}$ is equal to

If $\int \frac{\mathrm{d} x}{1+3 \sin ^2 x}=\frac{1}{2} \tan ^{-1}(\mathrm{f}(x))+\mathrm{c}$, where c is a constant of integration, then $\mathrm{f}(x)$ is equal to

\(\int \operatorname{cosec}(x-a) \cdot \operatorname{cosec} x d x=\)