Indefinite Integration · Mathematics · AP EAPCET

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

If $\int \frac{\log \left(1+x^4\right)}{x^3} d x=f(x) \log \left(\frac{1}{g(x)}\right)+\tan ^{-1}$ $(h(x))+c$, then $h(x)\left[f(x)+f\left(\frac{1}{x}\right)\right]=$

Let $f(x)=\int \frac{x}{\left(x^2+1\right)\left(x^2+3\right)} d x$. If $f(3)=\frac{1}{4} \log \left(\frac{5}{6}\right)$, then $f(0)=$

$$ \int \frac{2 \cos 2 x}{(1+\sin 2 x)(1+\cos 2 x)} d x= $$

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

$\int \frac{1}{x^5 \sqrt[3]{x^3+1}} d x=$

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

$\int\left(\tan ^9 x+\tan x\right) d x=0$

$\int \frac{\operatorname{cosec} x}{3 \cos x+4 \sin x} d x=$

$\int e^{2 x+3} \sin 6 x d x=$

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

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

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

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

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 $$\int \frac{d x}{x-3 y}=$$

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