NDA MATHEMATICS QUIZ

Question 1:

Value of the integral $\int \frac{x^{2} d x}{(x \cos x-\sin x)^{2}}$ will be

$\frac{x \sin x-\cos x}{x \cos x-\sin x}$

$\frac{x \sin x+\cos x}{x \cos x-\sin x}$

$\frac{x \sin x+\cos x}{x \sin x-\cos x}$

$\frac{\sin x-x \cos x}{x \sin x+\cos x}$

Question 2:

Value of the integral $\int \frac{x-1}{x+1} \frac{d x}{\sqrt{x\left(x^{2}+x+1\right)}}$ will be

$\tan ^{-1} \sqrt{\frac{x^{2}+x+1}{x}}$

$2 \tan ^{-1} \sqrt{\frac{x^{2}+x+1}{x}}$

$\tan ^{-1} \frac{\sqrt{x^{2}+x+1}}{x}$

$2 \tan ^{-1} \sqrt{x^{2}+x+1}$

Question 3:

$I=\int \frac{\sin x d x}{\sqrt{\left(a \cos ^{2} x+b \sin ^{2} x\right)}}$= . . . . . . .

$\frac{1}{\sqrt{b-a}} \cdot \cos ^{-1}\left(\sqrt{\frac{b-a}{b}} \cdot \cos x\right)$

$\frac{1}{\sqrt{b-a}} \cdot \sin ^{-1}\left(\sqrt{\frac{b-a}{b}} \cdot \cos x\right)$

$\frac{1}{\sqrt{b-a}} \cdot \sin ^{-1}\left(\sqrt{\frac{b-a}{b}} \cdot \sin x\right)$

$\frac{1}{\sqrt{b-a}} \cdot \cos ^{-1}\left(\sqrt{\frac{b-a}{b}} \cdot \sin x\right)$

Question 4:

$\int \frac{\left(x^{2}+1\right) d x}{x \sqrt{1+x^{4}}}$ equals

$\log \frac{x^{2}-1+\sqrt{x^{4}+1}}{x^{2}}$

$\log \frac{\left(x^{2}+1\right)+\sqrt{x^{4}+1}}{x}$

$\log \frac{\left(x^{2}+1\right)+\sqrt{x^{4}+1}}{x^{2}}$

$\log \frac{\left(x^{2}-1\right)+\sqrt{x^{4}+1}}{x}$

Question 5:

If $\int \frac{\cos x d x}{2 \sin x+3 \cos x}=c_{1} x+c_{2} f(x)$ then $f(x)$ equals

$\log (\sin x+\cos x)$

$\log (\sin x+3 \cos x)$

$\log (\sin x-\cos x)$

$\log (2 \sin x+3 \cos x)$

Question 6:

Value of the integral $\int(\log x)^{2} d x$ will be

$x(\log x)^{2}-2 x \log x+2 x$

$x^{2}(\log x)^{2}-2 x \log x+x$

$x(\log x)^{2}-2 x \log x+2 x^{2}$

$x(\log x)^{2}+2 x \log x-2 x^{2}$

Question 7:

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

$\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2}{\sqrt{3}} \tan \frac{x}{2}\right)$

$\frac{1}{\sqrt{3}} \tan ^{-1}\left(\tan \frac{x}{2}\right)$

$\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{1}{\sqrt{3}} \tan \frac{x}{2}\right)$

$\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{1}{\sqrt{3}} \tan \frac{x}{2}\right)$

Question 8:

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

$x(\sec x-\tan x)-\log (1+\sin x)$

$x(\sec x+\tan x)+\log (1-\sin x)$

$x(\sec x-\tan x)+\log (1-\sin x)$

$x(\sec x-\tan x)+\log (1+\sin x)$

Question 9:

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

$x \sec ^{-1} \sqrt{x}+\sqrt{x+1}+c$

$x^{2} \sec ^{-1} \sqrt{x}+\sqrt{x+1}+c$

$x \sec ^{-1} \sqrt{x}-\sqrt{x-1}+c$

$x^{2} \sec ^{-1} \sqrt{x}-\sqrt{x-1}+c$

Question 10:

If $\int \frac{x+1}{\sqrt{2 x-x^{2}}} d x=-f(x)+k \sin ^{-1}(x-1)$ then $\mathrm{k}=$ _________

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NDA MATHEMATICS QUIZ - 17

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