Indefinite Integration · Mathematics · MHT CET

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

If $$\int \frac{x^2}{\sqrt{1-x}} \mathrm{~d} x=\mathrm{p} \sqrt{1-x}\left(3 x^2+4 x+8\right)+\mathrm{c}$$ where $$\mathrm{c}$$ is a constant of integr...

$$\int \frac{\mathrm{d} x}{\cot ^2 x-1}=\frac{1}{\mathrm{~A}} \log |\sec 2 x+\tan 2 x|-\frac{x}{\mathrm{~B}}+\mathrm{c}$$, (where $$\mathrm{c}$$ is co...

If $$I=\int \frac{d x}{\sin (x-a) \sin (x-b)}$$, then I is given by

$$\int \frac{\sin 2 x\left(1-\frac{3}{2} \cos x\right)}{e^{\sin ^2 x+\cos ^3 x}} d x=$$

If $$\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^2 \theta} d \theta=A \log _e|f(\theta)|+c$$ (where $$c$$ is a constant of integration), then $$\f...

$$\int \frac{\sin x+\sin ^3 x}{\cos 2 x} d x=A \cos x+B \log \mathrm{f}(x)+c$$ (where $$\mathrm{c}$$ is a constant of integration). Then values of $$\...

If $$\int \frac{x^3 \mathrm{~d} x}{\sqrt{1+x^2}}=\mathrm{a}\left(1+x^2\right) \sqrt{1+x^2}+\mathrm{b} \sqrt{1+x^2}+\mathrm{c}$$ (where $$\mathrm{c}$$ ...

Let $$f(x)=\int \frac{x^2-3 x+2}{x^4+1} \mathrm{~d} x$$, then function decreases in the interval

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 $$g(2)$$...

$$\int \frac{x-3}{(x-1)^3} e^x d x=$$

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

If $$I=\int \frac{e^x}{e^{4 x}+e^{2 x}+1} d x$$ and $$J=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x$$, then for any arbitrary constant $$C$$, than the...

If $$\mathrm{I}=\int \frac{2 x-7}{\sqrt{3 x-2}} \mathrm{~d} x$$, then $$\mathrm{I}$$ is given by

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

If $$\int x^5 e^{-4 x^3} \mathrm{~d} x=\frac{1}{48} \mathrm{e}^{-4 x^3} \mathrm{f}(x)+\mathrm{c}$$, where $$\mathrm{c}$$ is a constant of integration,...

If $$\mathrm{f}(x)=\int \frac{x^2 \mathrm{~d} x}{\left(1+x^2\right)\left(1+\sqrt{1+x^2}\right)}$$ and $$\mathrm{f}(0)=0$$, then $$\mathrm{f}(1)$$ is...

$$\int \frac{1}{\cos ^3 x \sqrt{\sin 2 x}} d x=$$

If $$\int \frac{\sqrt{1-x^2}}{x^4} \mathrm{~d} x=\mathrm{A}(x)\left(\sqrt{1-x^2}\right)^{\mathrm{m}}+\mathrm{c}$$ for a suitable chosen integer $$\mat...

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

$$\int \frac{1}{(x+2)(1+x)^2} d x$$ has the value

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

The integral $$\int \frac{\sin ^2 x \cos ^2 x}{\left(\sin ^5 x+\cos ^3 x \sin ^2 x+\sin ^3 x \cos ^2 x+\cos ^5 x\right)^2} \mathrm{~d} x$$ is equal to...

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

If $$\int \cos ^{\frac{3}{5}} x \cdot \sin ^3 x d x=\frac{-1}{m} \cos ^m x+\frac{1}{n} \cos ^n x+c$$, (where $$\mathrm{c}$$ is the constant of integra...

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

If $$\int \frac{\cos 8 x+1}{\cot 2 x-\tan 2 x} \mathrm{~d} x=\mathrm{A} \cos 8 x+\mathrm{c}$$, where $$\mathrm{c}$$ is an arbitrary constant, then the...

The value of $$\int(1-\cos x) \cdot \operatorname{cosec}^2 x d x$$ is

If $$\mathrm{I}=\int \sin (\log (x)) \mathrm{d} x$$, then $$\mathrm{I}$$ is given by

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

If $$\int \frac{\mathrm{d} x}{x \sqrt{1-x^3}}=\mathrm{k} \log \left(\frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3}+1}\right)+\mathrm{c}$$, (where $$\mathrm{c}$$ i...

$$\int \frac{\log (\cot x)}{\sin 2 x} d x=$$

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

$$\int \frac{5 \tan x}{\tan x-2} \mathrm{~d} x=x+\mathrm{a} \log |\sin x-2 \cos x|+\mathrm{c},$$ (where $$c$$ is a constant of integration), then the ...

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

$$\int \mathrm{e}^x\left(1-\cot x+\cot ^2 x\right) \mathrm{d} x=$$

If $$\int \sqrt{\frac{x-7}{x-9}} d x=A \sqrt{x^2-16 x+63}+\log \left|(x-8)+\sqrt{x^2-16 x+63}\right|+c,$$ (where $$\mathrm{c}$$ is a constant of integ...

$$\int \frac{1}{7-6 x-x^2} d x=$$

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

If $$\mathrm{I}=\int \frac{\mathrm{d} x}{x^2\left(x^4+1\right)^{\frac{3}{4}}}$$, then $$\mathrm{I}$$ is

If $$\int \frac{\sin x}{3+4 \cos ^2 x} \mathrm{~d} x=\mathrm{A} \tan ^{-1}(\mathrm{~B} \cos x)+\mathrm{c}$$, (where $$\mathrm{c}$$ is a constant of in...

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

Let $$\alpha \in\left(0, \frac{\pi}{2}\right)$$ be fixed. If the integral $$\int \frac{\tan x+\tan \alpha}{\tan x-\tan \alpha} \mathrm{d} x=\mathrm{A}...

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

$$\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...

If $$ I=\int \frac{\sin x+\sin ^3 x}{\cos 2 x} d x=P \cos x+Q \log \left|\frac{\sqrt{2} \cos x-1}{\sqrt{2} \cos x+1}\right| $$ (where $$c$$ is a const...

$$\int \frac{1}{\sin (x-a) \sin x} d x=$$

If $$f(x)=\sqrt{\tan x}$$ and $$g(x)=\sin x \cdot \cos x$$ then $$\int \frac{f(x)}{g(x)} \mathrm{d} x$$ is equal to (where $$C$$ is a constant of inte...

$$\int \frac{3 x-2}{(x+1)(x-2)^2} \mathrm{~d} x=$$ (where $$C$$ is a constant of integration)

$$\int \frac{\sin \frac{5 x}{2}}{\sin \frac{x}{2}} d x=$$ (where $$C$$ is a constant of integration.)

$$\text { If } \int e^{x^2} \cdot x^3 \mathrm{~d} x=e^{x^2} \cdot[f(x)+C]$$ (where $$C$$ is a constant of integration.) and $$f(1)=0$$, then value of ...

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

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

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

$$\int \frac{\mathrm{dx}}{32-2 \mathrm{x}^2}=\mathrm{A} \log (4-\mathrm{x})+\mathrm{B} \log (4+\mathrm{x})+\mathrm{c}$$, then the values of $$\mathrm{...

$$\int \cos ^3 x \cdot e^{\log (\sin x)} d x=$$

If $$\int \frac{(\cos x-\sin x)}{8-\sin 2 x} d x=\frac{1}{p} \log \left[\frac{3+\sin x+\cos x}{3-\sin x-\cos x}\right]+c$$, then $$p=$$ (where $$\math...

$$\int \sec ^{-1} x d x=$$

If $$\int \frac{\sqrt{x}}{x(x+1)} d x=k \tan ^{-1} m+c$$, (where c is constant of integration), then

$$\int \frac{d x}{\cos x \sqrt{\cos 2 x}}=$$

If $$\int \frac{\sin x}{\sin (x-\alpha)} d x=A x+B \log \sin (x-\alpha)+c$$, then the value of A and B are respectively (where $$\mathrm{c}$$ is a con...

$$\int \frac{10^{\frac{x}{2}}}{\sqrt{10^{-x}-10^x}} d x=$$

$$\int e^{\left(e^x+x\right)} d x=$$

$$\int \frac{\tan ^4 \sqrt{x} \cdot \sec ^2 \sqrt{x}}{\sqrt{x}} d x=$$

$$\int \cos ^{-1} x d x=$$

$$\int \frac{1}{\cos x+\sqrt{3} \sin x} d x=$$

$$\int {{e^x}\left( {{{x - 1} \over {{x^2}}}} \right)dx = } $$

$$\int \sin ^{-1}\left(\frac{2 x}{1+x^2}\right) d x=\quad(\text { where }|x|

$$\int \frac{\sec ^8 x}{\operatorname{cosec} x} d x= $$

$$\int \frac{1}{x^{\frac{1}{2}}+x^{\frac{1}{3}}} d x=$$

$$\int[\sin |\log x|+\cos |\log x|] d x=$$

If $$\int {{{5\tan x} \over {\tan x - 2}}dx = x + a\log |\sin x - 2\cos x| + c} $$, then a = (Where c is constant of integration)

$$\int[1+2 \tan x(\tan x+\sec x)]^{\frac{1}{2}} d x= $$

If $$\int \frac{x^3}{\sqrt{1+x^2}} d x=a\left(1+x^2\right)^{\frac{3}{2}}+b \sqrt{1+x^2}+c$$, then $$a+b=$$, (where $$c$$ is constant of integration)...

$$\int e^{\tan x}\left(\sec ^2 x+\sec ^3 x \sin x\right) d x=$$

$$\int \sec ^4 x \cdot \tan ^4 x d x=\frac{\tan ^m x}{m}+\frac{\tan ^n x}{n}+c$$ (where c is constant of integration), then m + n =

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

$$\int \frac{2 x^2-1}{x^4-x^2-20} d x=$$

$$\int \tan ^{-1}(\sec x+\tan x) d x=$$

If $$\int \frac{1+x^2}{1+x^4} d x=\frac{1}{\sqrt{2}} \tan ^{-1}\left[\frac{f(x)}{\sqrt{2}}\right]+c$$, then $$f(x)=$$

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

$$\int \sin ^{-1} x d x=$$

$$\int \log x \cdot(\log x+2) d x=$$

$$\int \frac{d x}{x^2+4 x+13}=$$

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

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

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

If $$\int \frac{\sin \theta}{\sin 3 \theta} d \theta=\frac{1}{2 k} \log \left|\frac{k+\tan \theta}{k-\tan \theta}\right|+c$$, then $$k=$$

If $$\int \sqrt{x-\frac{1}{x}}\left(\frac{x^2+1}{x^2}\right) d x=\frac{2}{3}\left(x-\frac{1}{x}\right)^k+c$$, then value of $$k$$ is

$$\int \cot x \cdot \log [\log (\sin x)] d x=$$

$$\int \log x \cdot[\log (e x)]^{-2} d x=\ldots$$

If $\int \frac{1}{1-\cot x} d x=A x+B \log |\sin x-\cos x|+c$ then $A+B=\ldots \ldots$

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

If $$\int \frac{\cos x-\sin x}{8-\sin 2 x} d x=\frac{1}{p} \log \left[\frac{3+\sin x+\cos x}{3-\sin x-\cos x}\right]+c,$$ then $p=$ ................

$$\begin{aligned} & \text { If } \int \tan (x-\alpha) \tan (x+\alpha) \cdot \tan 2 x d x \\ & =p \log |\sec 2 x|+q \log |\sec (x+\alpha)| \\ & +r \log...

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

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

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

$$\int \frac{\sqrt{x^2-a^2}}{x} d x=\ldots \ldots$$