Indefinite Integration · Mathematics · AP EAPCET

If $\frac{x^4}{(x-1)(x-2)}=f(x)+\frac{A}{x-1}+\frac{B}{x-2}$, then $f(-2)+A+B=$

If $\int \frac{x^4+1}{x^2+1} d x=A x^3+B x^2+C x+D \tan ^{-1} x+E$, then $A+B+C+D=$

$$ \begin{aligned} & \text { If } \int \frac{x^2-x+2}{x^2+x+2} d x=x-\log (f(x))+\frac{2}{\sqrt{7}} \tan ^{-1}(g(x))+c, \text { then } \\ & f(-1)+\sqrt{7} g(-1)= \end{aligned} $$

$$ \int \sec \left(x-\frac{\pi}{3}\right) \sec \left(x+\frac{\pi}{6}\right) d x= $$

If $\int \frac{a \cos x+3 \sin x}{5 \cos x+2 \sin x} d x=\frac{26}{29} x-\frac{k}{29} \log |5 \cos x+2 \sin x|+\ldots$ then $|a+k|=$

If $\int \frac{d x}{1-\sin ^4 x}=A \tan x+B \tan ^{-1}(\sqrt{2} \tan x)+C$, then $A^2-B^2=$

If $\frac{x^2}{\left(x^2+2\right)\left(x^4-1\right)}=\frac{A}{x^2-1}+\frac{B}{x^2+1}+\frac{C}{x^2+2}$, then $A+B-C=$

If $\int \frac{5 \tan x}{\tan x-2} d x=a x+b \log |\sin x-2 \cos x|+c$, then $a+b=$

$$ \int x \cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right) d x(x>0)= $$

$$ \int \frac{d x}{(1+\sqrt{x}) \sqrt{x-x^2}}= $$

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

If $\int \frac{x}{x \tan x+1} d x=\log f(x)+k$, then $f\left(\frac{\pi}{4}\right)=$

If $\frac{2 x^4-3 x^2+4}{\left(x^2+1\right)\left(x^2+2\right)}=a+\frac{p x+q}{x^2+1}+\frac{m x+n}{x^2+2}$, then $\frac{n}{q}=$

$$ \int(\log 2 x)^3 d x= $$

$$ \int \frac{x+1}{(x-2) \sqrt{1-x}} d x= $$

$$ \int \frac{1}{1+x+x^2} d x= $$

If $\int \frac{d x}{(x \tan x+1)^2}=f(x)+C$, then $\lim\limits_{x \rightarrow \frac{\pi}{2}} f(x)=$

$$ \int \sin ^3 x \cos ^2 x d x= $$

19. If $\frac{3 x^3-7 x+1}{(x-2)^5}=\frac{A}{x-2}+\frac{B}{(x-2)^2}+\frac{C}{(x-2)^3}+\frac{D}{(x-2)^4}+\frac{E}{(x-2)^5}, \text { then } A(B+C+D+E)= $

$$ \int(\sqrt{\tan x}+\sqrt{\cot x}) d x= $$

$\int \frac{\sqrt{x-2}}{2 x+4} d x=$

If $\int x^{49}\left[\tan ^{-1} x^{50}+\frac{x^{50}}{1+x^{100}}\right] d x=\frac{x^n}{k} f(x)+c$, then

$$ f(x)-f\left(\sqrt[k]{x^n}\right)= $$

$$ \int \frac{x}{\sqrt{x^2-2 x+5}} d x= $$

For $0 < x < 1, \int\left[\tan ^{-1}\left(1-x+x^2\right)+\tan ^{-1}(1-x)\right] d x=$

If $\frac{3 x+1}{(x-1)\left(x^2+2\right)}=\frac{A}{x-1}+\frac{B x+C}{x^2+2}$, then $5(A-B)=$

$$ \int \frac{\sec ^2 x}{\sin ^7 x} d x-\int \frac{7}{\sin ^7 x} d x= $$

$$ \int \frac{d x}{(x+1) \sqrt{x^2+1}}= $$

If $\int \frac{d x}{2 \cos x+3 \sin x+4}=\frac{2}{\sqrt{3}} f(x)+C$, then $f\left(\frac{2 \pi}{3}\right)=$

If $\int \frac{1}{\left((x+4)^3(x+1)^5\right)^{1 / 4}} d x=A \cdot\left(\frac{x+4}{x+1}\right)^n+C$

$$ \int \frac{x+1}{x^3-1} d x= $$

$$ \int \frac{x^4-16 x^2+2 x+8}{x^3-4 x^2+2} d x= $$

$$ \int \frac{\sec ^2 x}{(\sec x+\tan x)^{\frac{5}{2}}} d x= $$

$$ \int \frac{1}{\cos x}\left[\frac{1}{\sin x}-\frac{1}{\sin x+3 \cos x}\right] d x= $$

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

If $\frac{a x+5}{\left(x^2+b\right)(x+3)}=\frac{x+21}{12\left(x^2+b\right)}+\frac{c}{12(x+3)}$, then $b^2=$

$$ \int\left(\sum_{r=0}^{\infty} \frac{x^r 2^r}{r!}\right) d x= $$

$$ \int \frac{d x}{12 \cos x+5 \sin x}= $$

If $\int \frac{\cos ^3 x}{\sin ^2 x+\sin ^4 x} d x=c-\operatorname{cosec} x-f(x)$, then $f\left(\frac{\pi}{2}\right)=$

$$ \int \frac{13 \cos 2 x-9 \sin 2 x}{3 \cos 2 x-4 \sin 2 x} d x= $$

$$ \int \sqrt{x^2+x+1} d x $$

If $\frac{x+1}{(x-1)^2\left(x^2+1\right)}=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C x+D}{x^2+1}$, then $\sqrt{3 A^2+4 D^2+5 C^2+B^2}=$

$$ \int \frac{1}{9 \cos ^2 x-24 \sin x \cos x+16 \sin ^2 x} d x= $$

If $\int \frac{1}{\cot \frac{x}{2} \cot \frac{x}{3} \cot \frac{x}{6}} d x=A \log \left|\cos \frac{x}{2}\right| +B \log \left|\cos \frac{x}{3}\right|+C \log \left|\cos \frac{x}{6}\right|+k$, then $A+B+C=$

$$ \int \frac{\sin x+\cos x}{\sin x-\cos x} d x= $$

$$ \int \frac{x^4-1}{x^2 \sqrt{x^4+x^2+1}} d x= $$

$$ \int \frac{(3 x-2) \tan \left(\sqrt{9 x^2-12 x+1}\right)}{\sqrt{9 x^2-12 x+1}} d x= $$

If $\int e^{\sin x}(1+\sec x \tan x) d x=e^{\sin x} f(x)+c$, then in $0 \leq x \leq 2 \pi$, then number of solutions of $f(x)=1$ is

If $\int \frac{d x}{(x-1)^{\frac{3}{2}}(x-3)^{\frac{1}{2}}}=\sqrt{f(x)}+C$, then $f(-1)-f(0)=$

$$ \int \frac{x}{\left(1-x^2\right) \sqrt{2-x^2}} d x= $$

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

If $\int x^2 \cos ^2 x d x=\frac{1}{6} f(x)+g(x) \sin 2 x +h(x) \cos 2 x+c$, then $f(1)+g(2)+h\left(\frac{1}{2}\right)=$

$$ \int \frac{e^{\sin x}(\sin 2 x-8 \cos x)}{2(\sin x-3)^2} d x= $$

If $\int\left(3 t^2 \sin \frac{1}{t}-t \cos \frac{1}{t}\right) d t=f(t) \sin \left(\frac{1}{t}\right)+C$ then $f(2)=$

$$ \int(\log x)^3 x^4 d x= $$

$$ \int \frac{\sin 2 x}{\sin ^2 x+3 \cos x-3} d x $$

If $\int \frac{d x}{\sin ^3 x+\cos ^3 x}=A \log \left|\frac{\sqrt{2}+t}{\sqrt{2}-t}\right|+B \tan ^{-1}(t)+C$, then $\left(\frac{B}{A}, t\right)=$

$\frac{4 x^2+5}{(x-2)^4}=\frac{A}{(x-2)}+\frac{B}{(x-2)^2}+\frac{C}{(x-2)^3}+\frac{D}{(x-2)^4}$, then $\sqrt{\frac{A}{C}+\frac{B}{C}+\frac{D}{C}}$ is equal to

$$ \text { If } \frac{13 x+43}{2 x^2+17 x+30}=\frac{A}{2 x+5}+\frac{B}{x+6} \text {, then } A+B \text { is equal to } $$

If $\int \log \left(6 \sin ^2 x+17 \sin x+12\right) \cos x d x=f(x)+c$, then $f\left(\frac{\pi}{2}\right)$ is equal to

$$ \int \frac{x^3 \tan ^{-1} x^4}{1+x^8} d x= $$

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

$$ \int \frac{\sin 7 x}{\sin 2 x \sin 5 x} d x= $$

If $\frac{1}{(3 x+1)(x-2)}=\frac{A}{3 x+1}+\frac{B}{x-2}$ and $\frac{x+1}{(3 x+1)(x-2)}=\frac{C}{3 x+1}+\frac{D}{x-2}$, then

$$ \begin{aligned} &\text { If } \int \frac{3}{2 \cos ^3 x \sqrt{2 \sin 2 x}} d x=\frac{3}{2}(\tan x)^B+\frac{3}{10}(\tan x)^A+C \text {, than }\\&A= \end{aligned} $$

$$\frac{2 x^2+1}{x^3-1}=\frac{A}{x-1}+\frac{B x+C}{x^2+x+1} \Rightarrow 7 A+2 B+C=$$

$$\int \frac{3 x+4}{x^3-2 x+4} d x=\log f(x)+C \Rightarrow f(3)=$$

$$\int \frac{e^{\tan ^{-1} x}}{1+x^2}\left[\left(\sec ^{-1} \sqrt{1+x^2}\right)^2+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right] d x=$$

$$\int \frac{d x}{(x-3)^{\frac{4}{5}}(x+1)^{\frac{6}{5}}}=$$

If $$I_n=\int\left(\cos ^n x+\sin ^n x\right) d x$$ and $$I_n-\frac{n-1}{n} I_{n-2} =\frac{\sin x \cos x}{n} f(x)$$, then $$f(x)=$$

If $$f(x)=\int x^2 \cos ^2 x\left(2 x \tan ^2 x-2 x-6 \tan x\right) d x$$ and $$f(0)=\pi$$, then $$f(x)=$$

If $$\int \frac{e^{\sqrt{x}}}{\sqrt{x}}(x+\sqrt{x}) d x=e^{\sqrt{x}}[A x+B \sqrt{x}+C]+K$$ then $$A+B+C=$$

If $$\int \frac{1+\sqrt{\tan x}}{\sin 2 x} d x=A \log \tan x+B \tan x+C$$, then $$4 A-2 B=$$

$$\int \frac{1+\tan x \tan (x+a)}{\tan x \tan (x+a)} d x=$$

Assertion (A) If $$I_n=\int \cot ^n x d x$$, then $$I_6+I_4=\frac{-\cot ^5 x}{5}$$

Reason (R) $$\int \cot ^n x d x=\frac{-\cot ^{n-1} x}{n} -\int \cot ^{n-2} x d x$$

If $$I_n=\int \tan ^n x d x$$, and $$I_0+I_1+2 I_2+2 I_3+2 I_4 +I_5+I_6=\sum_\limits{k=1}^n \frac{\tan ^k x}{k}$$, then $$n=$$

$$\int \frac{e^{\cot x}}{\sin ^2 x}(2 \log \operatorname{cosec} x+\sin 2 x) d x=$$

The parametric form of a curve is $$x=\frac{t^3}{t^2-1} y=\frac{t}{t^2-1}$$, then $$\int \frac{d x}{x-3 y}=$$

If

$$\begin{aligned} \frac{2 x^4-x^3+3 x^2-x+4}{x^2-3 x+2} =f(x)+\frac{A}{x-1}+\frac{B}{x-2}\end{aligned}$$, then

If $$f^{\prime}(x)=x+\frac{1}{x}$$, then $$f(x)$$ is equal to

If $$f(x)=\frac{1}{\left(\cos ^2 x\right) \sqrt{1+\tan x}}$$, then its antiderivative $$F(x)=$$ ........, given, $$F(0)=4$$

If the primitive of $$\cos (\log x)$$ is $$f(x)\{\cos (g(x))+\sin (h(x))\}$$, then which among the following is true?

$$\int \frac{\sec x}{\sqrt{\sin (2 x+\theta)+\sin \theta}} d x$$ is equal to

Given, $$\frac{3 x-2}{(x+1)^2(x+3)}=\frac{A}{x+1} +\frac{B}{(x+1)^2}+\frac{C}{x+3}$$, then $$4 A+2 B+4 C$$ is equal to

$$\int \frac{\sin \alpha}{\sqrt{1+\cos \alpha}} d \alpha$$ is equal to

If $$\int \frac{\cos 4 x+1}{\cot x-\tan x}=k \cos 4 x+C$$, then $$k$$ is equal to

If $$\int\left[\cos (x) \cdot \frac{d}{d x}(\operatorname{cosec}(x)] d x=f(x)+g(x)+c\right.$$ then $$f(x) \cdot g(x)$$ is equal to

If $$\int \frac{(2 x+1)^6}{(3 x+2)^8} d x=P\left(\frac{2 x+1}{3 x+2}\right)^Q+R$$, then $$\frac{P}{Q}$$ is equal to

Which of the following is partial fraction of $$\frac{-x^2+6 x+13}{(3 x+5)\left(x^2+4 x+4\right)}$$ is equal to

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

$$\int(\cos x) \log \cot \left(\frac{x}{2}\right) d x$$ is equal to

$$\int \sqrt{e^{4 x}+e^{2 x}} d x$$ is equal to

If $$\int \frac{1}{1+\sin x} d x=\tan (f(x))+c$$, then $$f^{\prime}(0)$$ is equal to

$$\int \frac{e^x(x+3)}{(x+5)^3} d x$$ is equal to

If $$\int \frac{(x-1)^2}{\left(x^2+1\right)^2} d x=\tan ^{-1}(x)+g(x)+k$$, then $$g(x)$$ is equal to

If $$\int \frac{1-(\cot x)^{2021}}{\tan x+(\cot x)^{2022}} d x=\frac{1}{A} \log\left|(\sin x)^{2023}+(\cos x)^{2023}\right|+c$$, then $$A$$ is equal to