NDA MATHEMATICS SECTIONAL TEST 3

Question 1:

$\int \frac{d \theta}{\sqrt{1+\sin \theta}}=\ldots \ldots$

$\sqrt{2} \log \tan \left(\frac{\pi}{8}+\frac{\theta}{2}\right)$

$\sqrt{3} \log \tan \left(\frac{\pi}{8}+\frac{\theta}{4}\right)$

$\frac{1}{\sqrt{2}} \log \tan \left(\frac{\pi}{8}+\frac{\theta}{4}\right)$

$\sqrt{3} \log \tan \left(\frac{\pi}{8}+\frac{\theta}{6}\right)$

Question 2:

Value of the integral$\int \frac{\sin x d x}{\sqrt{\cos ^{2} x+2 \cos x+2}} \text { is }$

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

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

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

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

Question 3:

$\int \frac{\sec ^{4} \theta \cdot d \theta}{\tan \theta}=$. . . . . . . .

$\log \tan \theta+\tan ^{2} \theta$,

$\quad \log \tan \theta+\frac{1}{2} \tan ^{2} \theta$

$2 \log \tan \theta+3 \tan \theta$

$\quad \log \tan \theta+\frac{1}{3} \tan ^{3} \theta$

Question 4:

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

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

$\frac{1}{\sqrt{3}} \tan ^{-1} \frac{1+2 x^{2}}{\sqrt{3}}$

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

$\frac{1}{\sqrt{5}} \cdot \tan ^{-1} \frac{1-2 x^{2}}{3}$

Question 5:

Value of the indefinite integral $\int \frac{\left(x^{4}-1\right) \cdot d x}{x^{2} \sqrt{x^{4}+x^{2}+1}}$ is

$\frac{1}{x} \cdot \sqrt{x^{4}+2 x^{2}+3}$

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

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

None of these

Question 6:

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

$\frac{\sqrt{z}-1}{\sqrt{1-z}}+c$

$\frac{2(\sqrt{z}-1)}{\sqrt{1-z}}+c$

$\frac{2(\sqrt{z}+1)}{\sqrt{1-z}}+c$

$\frac{2(\sqrt{z}-z)}{\sqrt{1+z}}+c$

Question 7:

Value of the integral $\int \frac{x^{2}}{\sqrt{1+x}} \cdot d x$ will be

$\sqrt{1+x}\left\{\frac{(1+x)^{2}}{5}-\frac{2}{3}(1+x)+1\right\}$

$2 \sqrt{1+x}\left\{\frac{(1+x)^{2}}{5}-\frac{2}{3}(1+x)+1\right\}$

$\left\{\frac{(1+x)^{\frac{5}{2}}}{5}-\frac{2}{3}(1+x)^{\frac{1}{2}}+1\right\}$

$\frac{2}{5}\left[(1+x)^{\frac{5}{2}}-(1+x)^{\frac{3}{2}}+1\right]$

Question 8:

$\int \sqrt{\frac{1-x}{1+x}} \cdot d x$ equals

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

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

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

$\sin ^{-1} x+\sqrt{1-x^{2}}+C$

Question 9:

Value of the integral$\int \frac{e^{2 x}}{\left(1+e^{x}\right)} \cdot d x$ is.

$e^{x}-\log \left(1+e^{x}\right)+c$

$e^{x}+\log \left(1+e^{x}\right)+c$

$x \log \left(1+e^{x}\right)+c$

$\frac{x}{\log \left(1+e^{x}\right)}+c$

Question 10:

Value of the integral$\int \frac{\sin 2 x \cdot d x}{\left(a^{2} \cos ^{2} x+b^{2} \sin ^{2} x\right)^{2}}\text { is }$

$-\frac{1}{\left(a^{2}+b^{2}\right)}\left(a^{2} \cos ^{2} x+b^{2} \sin ^{2} x\right)^{-2}$

$\frac{1}{b^{2}-a^{2}}\left(a^{2} \sin ^{2} x+b^{2} \cos ^{2} x\right)^{-1}$

$-\frac{1}{b^{2}-a^{2}}\left(a^{2} \cos ^{2} x+b^{2} \cos ^{2} x\right)^{-1}$

$\frac{1}{a^{2}+b^{2}}\left(a^{2} \sin ^{2} x-b^{2} \cos ^{2} x\right)^{-2}$

Question 11:

If $\mathrm{I}=\int \frac{\sqrt{x^{2}-a^{2}}}{x} d x$ then value of $\mathrm{I}$ will be

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

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

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

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

Question 12:

$\int \frac{\sec ^{2} x \cdot \tan x}{\sec ^{2} x-1} \cdot d x=\ldots \ldots \ldots$

$\log \cot x$

$\log (\sec x \cdot \tan x)$

$\log \tan x$

$\log \sin x$

Question 13:

If $\int \frac{d x}{x \sqrt{1+x^{2}}}=k\log \{f(x)\} \quad$ where $f(x)=\frac{1+\sqrt{x^{2}+1}}{x}$ then $\mathrm{k}$ equals

$-1$

2.0

$\frac{1}{2}$

Question 14:

$\int \frac{d z}{\sqrt{z}+\sqrt{1+z}}=\ldots .$

$\frac{2}{3}\left[(z+1)^{\frac{3}{2}}-z^{\frac{3}{2}}\right]$

$\frac{2}{5}\left[(z+1)^{\frac{1}{2}}-z\right]$

$\frac{2}{7}\left[(z+1)^{\frac{3}{2}}-z^{2}\right]$

$\frac{1}{5}\left[(z+1)^{\frac{3}{2}}-z^{\frac{5}{2}}\right]$

Question 15:

Value of the integral $\int \frac{1-x^{2}}{\left(1+x^{2}\right)} \cdot \frac{d x}{\sqrt{1+x^{4}}}$ is

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

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

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

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

Question 16:

If $\int \frac{d x}{\left(1+x^{2}\right)^{\frac{3}{2}}}-f(x)+\mathrm{c}$; then $f(x)$ equals

$\frac{x}{\left(1+x^{2}\right)^{\frac{3}{2}}}$

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

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

$\frac{x^{2}}{\left(1+x^{2}\right)^{\frac{3}{2}}}$

Question 17:

If $\mathrm{I}=\int \frac{(t+1)^{2}}{t\left(1+t^{2}\right)} \cdot d t$ then $\mathrm{I}$ equals.

$2 \tan ^{-1} t-t+c$

$\tan ^{-1} t-\frac{1}{2} t^{2}+c$

$\left(\tan ^{–1} t\right)^{2}-\frac{t^{3}}{3}+c$

$\log t+2 \tan ^{-1} t+c$

Question 18:

$\int \frac{d x}{\sqrt{(x-\alpha)(x-\beta)}}=\ldots \ldots .$

$\log (\sqrt{x-\alpha}+\sqrt{x-\beta})$

$2 \log (\sqrt{x-\alpha}+\sqrt{x-\beta})$

$-\log (\sqrt{x-\alpha}+\sqrt{x-\beta})$

$\log (\sqrt{x-\alpha}-\sqrt{x-\beta})$

Question 19:

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

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

$x+\frac{1}{2} \log (\sin x+\cos x)$

$2 x^{2}+\log (\sin x+\cos x)^{2}$

$\frac{1}{2}\{x+\log (\sin x+\cos x)\}$

Question 20:

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

$2 \tan ^{2} \frac{x}{2}-x$

$2 \tan \frac{x}{2}-x^{2}$

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

$2 \tan \frac{x}{2}-x$