Improper Integrals

Q. 40.Express in terms of the ratios of a positive angle, which is less than 45^° , the quantities(a) sin 168"

Q.

If A+B +C =${180}^{\circ }$, then $\sum tan\frac{A}{2}tan\frac{B}{2}$ is equal

1. 3

2. 0

3. 2

4. 1

Q. find the value of tan 19π /3 (with explanation!

Q.

cosA + cos (120° + A) + cos(120° - A) =

1. 0

2. 1

3. -1/2

4. 1/2

Q. More than One Answer Type
एक से अधिक उत्तर प्रकार के प्रश्न

Value of $\int \left(\frac{2x}{x^4+x^2+1}\right)dx$ equals to

$\int \left(\frac{2x}{x^4+x^2+1}\right)dx$ का मान है

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

Q. $\int \frac{x^{7}dx}{\sqrt{1+x^4}}$equals

$\int \frac{x^{7}dx}{\sqrt{1+x^4}}$ का मान है

1. $\frac{1}{6}\sqrt{1+x^4}\left(x^4-2\right)+c$
2. $\frac{1}{3}\sqrt{1+x^4}\left(x^4-2\right)+c$
3. $\frac{\sqrt{1+x^4}}{6}\left(x^4-11\right)+c$
4. $\frac{\sqrt{1+x^4}}{6}\left(x^4-12\right)+c$

Q. $\mathrm{If}{\int }_0^{\infty }\frac{\sin x}{x}\mathrm{dx}=\frac{\pi }{2},\mathrm{then}{\int }_0^{\infty }\frac{{\sin }^{3}x}{x}\mathrm{dx}$
is equal to

1. $\frac{\pi }{2}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{4}$
4. $\frac{3\pi }{2}$

Q. The range of the functions $f\left(x\right)={\int }_1^x|t|dt,xϵ[\frac{-1}{2},\frac{1}{2}]$ is

1. $[\frac{3}{8},\frac{5}{8}]$
2. $[\frac{-5}{8},\frac{3}{8}]$
3. $[\frac{-3}{8},\frac{5}{8}]$
4. $[\frac{-5}{8},\frac{-3}{8}]$

Q. Prove the following identities (1-17)

$\frac{\cos \mathrm{\theta }}{1-\sin \mathrm{\theta }}=\frac{1+\cos \mathrm{\theta }+\sin \mathrm{\theta }}{1+\cos \mathrm{\theta }-\sin \mathrm{\theta }}$

Q. ${\int }_{-4}^4|x+2|dx=$

1. 50
2. 24
3. 20

4. None of these

Q. ${\int }_{-1}^3(Ta{n}^{-1}\frac{x}{x^2+1}+Ta{n}^{-1}\frac{x^2+1}{x})dx=$

1. $\pi$
2. $2\pi$
3. $4\pi$
4. $none$

Q. Express sin 10 theta in terms of functions of 5 theta

Q. Suppose f, g are continuous and differentiable on [0, b], are non-negative on [0, b] and f is non constant with f(0) = 0, then the minimum value of ${\int }_0^{a}g\left(x\right)f^{\prime }\left(x\right)dx+{\int }_0^b{g}^{\prime }\left(x\right)f\left(x\right)dxonaϵ(0,b]$ is

1. f(a).g(a)
2. f(b).g(b)
3. f(a).g(b)
4. f(b).g(a)

Q. 6. tan 2A = cot (A-18^° ) if A is an acute angle .