50 questions · timed · auto-graded
$\int\text{x log 2x dx}$.
= $\frac{\text{x}^{2}}{2}\cdot\log\text{2x}-\frac{1}{2}\int\text{x dx + c}_{1}=\frac{\text{x}^{2}}{2}\cdot\log\text{ 2x}-\frac{\text{x}^{2}}{4}+\text{c}.$
$\int\sin\text{4x}\cos\text{3x dx}$.
$=\frac{1}{2}\Bigg[\frac{-\cos\text{ 7x}}{7}-\cos\text{x}\Bigg]+\text{c }\text{ OR }-\frac{1}{14}\cos\text{ 7x}-\frac{1}{2}\cos\text{ x + c}$.
$\int\frac{\text{2x.tan-1(x}^{2})}{\text{1 + x}^{4}}\text{dx}. $
$\int \frac{1 + x^{2}}{1 + x^{4}} \text{dx}$
$= \frac{1}{\sqrt{2}} \tan^{-1} \bigg(x-\frac{\frac{1}{x}}{\sqrt{2}}\bigg) + c $
$= \frac{1}{\sqrt{2}} \tan^{-1}\bigg(\frac{x^{2} -1}{\sqrt{2x}}\bigg) + c $
$= \frac{1}{2} \bigg[ \frac{\sin 7x}{7} + \sin x\bigg] + c$or $\frac{1}{14} \sin 7 x + \frac{1}{2} \sin x + c$
$\int{\frac{dx} {\sqrt{x^{2} - 3x + 2}}}$
$= \log\bigg| \bigg(x - \frac{3}{2}\bigg) + \sqrt{x^{2} - 3x + 2}\bigg| + c$
$\int \frac{\sin (\text{x} - \alpha)}{\sin (\text{x} + \alpha)} \text{dx}$
$= \int \frac{\sin(x + \alpha). \cos 2\alpha - \cos (x + \alpha) .2 \alpha}{\sin(x + \alpha)} dx$
$= \cos 2 \alpha \int dx - \sin 2 \alpha \int \frac{\cos(x + \alpha)}{\sin(x + \alpha)} dx$
$= x \cos 2 \alpha - \sin 2 \alpha \log |\sin (x + \alpha)|+c$
$\frac{\text{x}^{2}+\text{x}+1}{(\text{x}+1)^{2}(\text{x}+2)}$
$\int\text{x}^3\cos\text{x}^2\text{dx}$
Let
$\sin2\text{t}=\text{u}$ Then, $2\cos2\text{t dt} =\text{du}$When
$\text{t}=0,\text{u}=0$ and $\text{t}=\frac{\pi}{4},\text{u}=1$$\therefore\ \text{I}=\frac{1}{2}\int^\limits{1}_{0}\text{u}^3\text{ du}$
$\Rightarrow\text{I}=\frac{1}{2}\Big[\frac{\text{u}^4}{4}\Big]^1_0$
$\Rightarrow\text{I}=\frac{1}{2}\Big(\frac{1}{4}-0\Big)$
$\Rightarrow\text{I}=\frac{1}{8}$
Putting x + 2 = t
⇒ x = t - 2
$\frac{\text{dx}}{\text{dt}}=1$
⇒ dx = dt
Putting this value in eq. (i),
$\text{I}=\int\frac{3(\text{t}-2)-1}{(\text{t})^2}\text{dt}=\int\frac{3\text{t}-6-1}{\text{t}^2}\text{dt}=\int\frac{3\text{t}-7}{\text{t}^2}\text{dt}$
$=\int\Bigg(\frac{3\text{t}}{\text{t}^2}-\frac{7}{\text{t}^2}\Bigg)\text{dt}=\int\Bigg(\frac{3}{\text{t}}-\frac{7}{\text{t}^2}\Bigg)\text{dt}=3\int\frac{1}{\text{t}}\text{dt}-7\int\text{t}^{-2}\text{dt}$
$=3\text{log}|\text{t}|-7\frac{\text{t}^{-1}}{-1}+\text{c}=3\text{log}|\text{t}|+\frac{7}{\text{t}}+\text{c}=3\text{log}|\text{x}+2|+\frac{7}{\text{x}+2}+\text{c}$
$\int\frac{1}{\sqrt{2\sin^2\text{x}}}\text{dx}\ \big[\because 1-\cos 2\text{x}=2\sin^2\text{x}\big]$
$=\frac{1}{\sqrt{2}}\int\text{cosec x dx}$
$=\frac{1}{\sqrt{2}}\text{ln}|\text{cosec x}-\cot\text{x}|=\text{C}$
$=\frac{1}{\sqrt{2}}\text{ln}\Big|\frac{1}{\sin\text{x}}-\frac{\cos\text{x}}{\sin\text{x}}\Big|+\text{C}$
$=\frac{1}{\sqrt{2}}\text{ln}\bigg|\frac{2\sin^2\frac{\text{x}}{2}}{\sin\text{x}}\bigg|+\text{C} \Big[\because 1-\cos\text{x}=2\sin^2\frac{\text{x}}{2}\Big]$
$=\frac{1}{\sqrt{2}}\text{ln}\Bigg|\frac{2\sin^2\frac{\text{x}}{2}}{2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}\Bigg|+\text{C}\ \Big[\because\sin\text{x}=2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}\Big]$
$=\frac{1}{\sqrt{2}}\text{ln}\Big|\tan\frac{\text{x}}{2}\Big|+\text{C}$
$=\int\frac{\cos^2\text{x}-\sin^2\text{x}}{(\cos\text{x}+\sin\text{x})^2}\text{dx}$
$=\int\frac{\cos\text{x}-\sin\text{x}}{\cos\text{x}+\sin\text{x}}\text{dx}$
Putting
$\cos\text{x}+\sin\text{x}=\text{t}$$\Rightarrow-\sin\text{x}+\cos\text{x}=\frac{\text{dt}}{\text{dt}}$
$\Rightarrow(\cos\text{x}-\sin\text{x})\text{dx}=\text{dt}$
$\therefore\text{I}=\int\frac{1}{\text{t}}\text{dt}$
$=\text{ln}|\text{t}|+\text{C}$
$=\text{ln}|\cos\text{x}+\sin\text{x}|+\text{C}\ \big[\because\text{t}=\cos\text{x}+\sin\text{x}\big]$
$\int\frac{\cos\text{x}}{\sqrt{\sin^2\text{x}-2\sin\text{x}-3}}\text{ dx}$
$\int\frac{\sin\text{x}-\cos\text{x}}{\sqrt{\sin2\text{x}}}\text{ dx}$
$\int\frac{\text{x}\cos^{-1}\text{x}}{\sqrt{1-\text{x}^3}}\text{dx}$
$\int\text{e}^{\text{x}}\big(\cot\text{x}-\text{cosec}^2\text{x}\big)\text{dx}$
$\int\frac{1}{\text{x}\sqrt{4-9(\log\text{x})^2}}\text{ dx}$
$\int\frac{\text{x}+\sin\text{x}}{1+\cos\text{x}}\text{dx}$
$\int\log_{10}\text{x dx}$
$\int\frac{1}{\sqrt{16-6\text{x}-\text{x}^2}}\text{ dx}$
$\int\frac{\text{x}+7}{3\text{x}^2+25\text{x}+28}\text{ dx}$
$=\frac{1}{3}\ln|3\text{x}+4|+\text{C}$
$\int\text{e}^{\text{x}}(\tan\text{x}-\log\cos\text{x})\text{dx}$
$=\int\text{e}^{\text{x}}\tan\text{x dx}-\int\text{e}^{\text{x}}\log\cos\text{x dx}$
Integrating by parts
$=\int\text{e}^{\text{x}}\tan\text{x dx}-\Big\{\text{e}^{\text{x}}\log\cos\text{x}-\int\text{e}^{\text{x}}\Big(\frac{\text{d}}{\text{dx}}\log\cos\text{x}\Big)\text{dx}\Big\}$
$=\int\text{e}^{\text{x}}\tan\text{x dx}-\big\{\text{e}^{\text{x}}\log\cos\text{x}+\int\text{e}^{\text{x}}\tan\text{x dx}\big\}$
$=\int\text{e}^{\text{x}}\tan\text{x dx}-\text{e}^{\text{x}}\log\cos\text{x}-\int\text{e}^{\text{x}}\tan\text{x dx}+\text{C}$
$=-\text{e}^{\text{x}}\log\cos\text{x}+\text{C}$
$=\text{e}^{\text{x}}\log\sec\text{x}+\text{C}$ $\big[\because\log\sec\text{x}=-\log\cos\text{x}\big]$
$\int\frac{1}{\text{x}^{\frac{2}{3}}\sqrt{\text{x}^{\frac{2}{3}}-4}}\text{ dx}$
$\int\text{e}^{\text{x}}\big[\sec\text{x}+\log(\sec\text{x}+\tan\text{x})\big]\text{dx}$
$\int\text{e}^{\text{}x}\Big(\frac{1+\sin\text{x}}{1+\cos\text{x}}\Big)\text{dx}$
$\int\frac{\cos2\text{x}}{\sqrt{\sin^22\text{x}+8}}\text{ dx}$
$\int\frac{1}{\sqrt{(1-\text{x}^2)\big\{9+\big(\sin^{-1}\text{x}\big)^2\big\}}}\text{ dx}.$
$\int\frac{\sin2\text{x}}{\sqrt{\cos^4\text{x}-\sin^2\text{x}+2}}\text{ dx}$
$\int\text{e}^{\text{x}}\sec\text{x}(1+\tan\text{x})\text{dx}$
$ \int^{\frac{\pi}{2}}_{\frac{-\pi}{2}}\sin^{2}\text{x}\ \text{dx}$
$\Rightarrow\ \ \text{I}=2\int\limits_{0}^{\frac{\pi}{2}}\sin^{2}\bigg(\frac{\pi}{2}-\text{x}\bigg)\text{dx}\ \ \ \Big[\because\int^{\text{a}}\limits_{0}\text{f}\text{(x)}\ \text{dx}=\int\limits_{0}^{\text{a}}\text{f}\text{(a}-\text{x)}\text{dx}=\Big]$
$\Rightarrow\ \ \text{I}=2\int^{\frac{\pi}{2}}\limits_{0}\cos^{2}\text{x}\ \text{dx}$
Adding eq. (i) and (ii),
$21=2\int^{\frac{\pi}{2}}\limits_{0}\big(\sin^{2}\text{x}+\cos^{2}\text{x}\big)\text{dx}=2\int^{\frac{\pi}{2}}\limits_{0}1\text{dx}=2\text{(x)}^{\frac{\pi}{2}}_{0}=2.\frac{\pi}{2}=\pi$
$\Rightarrow\ \ \ \text{I}=\frac{\pi}{2}$
$\int\text{x}\sin\text{x}\cos\text{x dx}$
$\int2\text{x}^3\text{e}^{\text{x}^{2}}\text{dx}$
$\int\frac{\log(\log\text{x})}{\text{x}}\text{dx}$
$\int\frac{\text{x}+5}{3\text{x}^2+13\text{x}-10}\text{ dx}$
$\int\sec^{-1}\sqrt{\text{x}}\text{dx}$
$\int\text{e}^{\text{x}}(\log\text{x}+\frac{1}{2})\text{dx}$