Integration of Trigonometric Functions

Q. The value of the integral $\int \frac{x^4+1}{x^6+1}dx$ is
$(C$ is a constant of integration$)$

1. ${\tan }^{-1}x+3{\tan }^{-1}{x}^3+C$
2. $\frac{1}{3}{\tan }^{-1}x+{\tan }^{-1}{x}^3+C$
3. $\frac{1}{3}{\tan }^{-1}x+3{\tan }^{-1}{x}^3+C$
4. ${\tan }^{-1}x+\frac{1}{3}{\tan }^{-1}{x}^3+C$

Q. $\int \frac{x^2+{\cos }^{2}x}{x^2+1}{\text{cosec}}^{2}xdx$ is equal to

1. $\cot x-{\cot }^{-1}x+c$
2. $c-\cot x+{\cot }^{-1}x$
3. ${\tan }^{-1}x-\frac{\text{cosec}x}{\sec x}+c$
4. $-e^{\log {\tan }^{-1}x}-\cot x+c$

Q. $\int \frac{\text{dx}}{{\text{sin}}^4\text{x}+{\text{cos}}^4\text{x}}$ is equal to

1. $\frac{1}{\sqrt{2}}{\text{tan}}^{-1}\left(\frac{1}{\sqrt{2}}\text{tan}2x\right)+\text{C}$
2. $\sqrt{2}{\text{tan}}^{-1}\left(\frac{1}{\sqrt{2}}\text{tan}2x\right)+\text{C}$
3. $\frac{1}{\sqrt{2}}{\text{tan}}^{-1}\left(\frac{1}{\sqrt{2}}\text{cot}2\text{x}\right)+\text{C}$
4. None of these

Q. If $\int \frac{x^4+1}{x^6+1}dx={\tan }^{-1}f\left(x\right)-\frac{2}{3}{\tan }^{-1}g\left(x\right)+c,$ where $c$ is an arbitrary constant, then

1. both $f\left(x\right)$ and $g\left(x\right)$ are odd functions
2. both $f\left(x\right)$ and $g\left(x\right)$ are even functions
3. $f\left(x\right)=g\left(x\right)$ has no real roots
4. $\int \frac{f\left(x\right)}{g\left(x\right)}dx=\frac{-1}{x}+\frac{1}{3x^3}+d,$ where $d$ is an arbitrary constant.

Q.

Solve the given limit problem $\underset{x\to 1}{\lim }\frac{{\int }_0^{{\left(x-1\right)}^2}t\cos \left(t^2\right)dt}{\left(x-1\right)\sin \left(x-1\right)}$.

1. is equal to $1$

2. is equal to $\frac{1}{2}$

3. $0$

4. is equal to $-\frac{1}{2}$

Q.

Prove $x=\frac{n\pi }{2}$ or $x=(\frac{m\pi }{2}+\frac{3\pi }{8})$, where m, $n\in I$

Q. If ${\theta }_1$ and ${\theta }_2$ be respectively the smallest and the largest values of $\theta$ in $(0,2\pi )-\left\{\pi \right\}$ which satisfy the equation, $2{\cot }^2\theta -\frac{5}{\sin \theta }+4=0$, then $\underset{{\theta }_1}{\overset{{\theta }_2}{\int }}{\cos }^{2}3\theta d\theta$ is equal to :

1. $\frac{2\pi }{3}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{3}+\frac{1}{6}$
4. $\frac{\pi }{9}$

Q. If $\Phi \left(x\right)=\int \frac{dx}{si{n}^{\frac{1}{2}}xco{s}^{\frac{7}{2}}x}$, then $\Phi \left(\frac{\pi }{4}\right)-\Phi \left(0\right)=$

1. $\frac{12}{5}$
2. $\frac{9}{5}$
3. $\frac{6}{5}$
4. \N

Q. The value of integral $\int \sqrt{\frac{x-2}{x-3}}dx$ is equal to
(Where $C$ is integration constant)

1. $-\sqrt{(3-x)(2-x)}+\ln |\sqrt{3-x}+\sqrt{2-x}|+C$
2. $\frac{1}{2}\sqrt{(3-x)(2-x)}-\frac{1}{2}\ln |\sqrt{3-x}+\sqrt{2-x}|+C$
3. $-\frac{1}{2}\sqrt{(3-x)(2-x)}-\ln |\sqrt{3-x}+\sqrt{2-x}|+C$
4. $-\frac{1}{2}\sqrt{(3-x)(2-x)}-\frac{1}{2}\ln |\sqrt{3-x}+\sqrt{2-x}|+C$

Q.

$\int \frac{1}{2+3sinx}dx$

1. $\frac{2}{\sqrt{5}}.log\left|\frac{tan\frac{x}{2}+\frac{3}{2}-\sqrt{\frac{5}{4}}}{tan\frac{x}{2}+\frac{3}{2}+\sqrt{\frac{5}{4}}}\right|+C$

2. $\frac{1}{2\sqrt{5}}.log\left|\frac{tan\frac{x}{2}+\frac{3}{2}-\sqrt{\frac{5}{4}}}{tan\frac{x}{2}+\frac{3}{2}+\sqrt{\frac{5}{4}}}\right|+C$

3. $\frac{3}{\sqrt{5}}.log\left|\frac{tan\frac{x}{2}+\frac{3}{2}-\sqrt{\frac{5}{4}}}{tan\frac{x}{2}+\frac{3}{2}+\sqrt{\frac{5}{4}}}\right|+C$

4. $\frac{1}{\sqrt{5}}.log\left|\frac{tan\frac{x}{2}+\frac{3}{2}-\sqrt{\frac{5}{4}}}{tan\frac{x}{2}+\frac{3}{2}+\sqrt{\frac{5}{4}}}\right|+C$

Q. $\int \frac{\mathrm{dx}}{4+5{\sin }^{2}x}\mathrm{dx}$ is equal to.

1. 
   $\frac{1}{6}{\tan }^{-1}\left(\frac{3\mathrm{cotx}}{2}\right)+C$
2. 
   $\frac{1}{6}{\tan }^{-1}\left(\frac{3\mathrm{tanx}}{2}\right)+C$
3. 
   $\frac{1}{6}{\cot }^{-1}\left(\frac{3\mathrm{cotx}}{2}\right)+C$
4. 
   $\frac{1}{6}{\cot }^{-1}\left(\frac{3\mathrm{tanx}}{2}\right)+C$

Q. $\int \frac{x+sinx}{1+cosx}dx=$

1. $log|1+cosx|+C$
2. $xtan\frac{x}{2}+C$
3. $-log|1+cosx|+C$
4. $xcot\left(\frac{x}{2}\right)+C$

Q. STATEMENT - 1 : $cos{15}^{\circ }=\frac{\sqrt{3}+1}{2\sqrt{2}}$
STATEMENT - 2 : $cos(A-B)=cosAcosB+sinAsinB$

1. Statement - 1 is True, Statement - 2 is True, Statement - 2 is a correct explanation for Statement - 1
2. Statement - 1 is True, Statement - 2 is False
3. Statement - 1 is False, Statement - 2 is True
4. Statement - 1 is True, Statement - 2 is True : Statement 2 is NOT a correct explanation for Statement - 1

Q. Evaluate $\int \frac{1}{\sqrt{3}\sin x+\cos x}dx$.

1. $\frac{1}{2}\log \left|\tan (\frac{x}{2}+\frac{\pi }{12})\right|+c$
2. $\frac{1}{2}\log \left|\tan (\frac{x}{2}+\frac{\pi }{8})\right|+c$
3. $\frac{1}{3}\log \left|\tan (\frac{x}{2}+\frac{\pi }{12})\right|+c$
4. none of these

Q. If $\int \frac{cos8x+1}{tan2x-cot2x}dx=acos8x+C$, then

1. $a=\frac{-1}{16}$
2. $a=\frac{1}{8}$
3. $a=\frac{1}{16}$
4. $a=\frac{-1}{8}$

Q. If $\int \frac{x\cot x+1}{\sqrt{cose{c}^{2}x-x^2}}dx=f\left(x\right)+c,$ where $\underset{x\to 0^{+}}{lim}f\left(x\right)=0$ and $\text{cosec}x>0$. (where $c$ is a constant of integration), then which of the following is/are incorrect

1. $f\left(1\right)=1$
2. $\underset{x\to 0}{lim}\frac{f\left(x\right)}{|x|\tan x}$ does not exist.
3. $f\left(1\right)=\frac{\pi }{2}-1$
4. $f\left(x\right)$ is periodic with a period $2\pi$

Q. If $\int \sqrt{1+\sin x}f\left(x\right)dx=\frac{2}{3}(1+\sin x{)}^{3/2}+c,$ then $f\left(x\right)$ equals

1. $\cos x$
2. $\sin x$
3. $\tan x$
4. $1$

Q. $\int \frac{co{s}^{4}xdx}{si{n}^{3}x(si{n}^{5}x+co{s}^{5}x{)}^{\frac{3}{5}}}=-\frac{1}{2}(1+co{t}^{A}x{)}^B+C$ then AB=

1. 1
2. 2
3. $\frac{1}{2}$
4. None of these

Q. If $I=\int (\sec x-\sin x\tan x)dx,$ then the value of $I$ is
(where $C$ is constant of integration)

Q. If $f\left(x\right)=\int e^x(\frac{2}{1-\tan x}+{\tan }^2(x+\frac{\pi }{4}\left)\right)dx$, where $f\left(\frac{3\pi }{4}\right)=0.$ Then the value of $\ln \left(f\right(\pi \left)\right)$ is

Q. If $\int \sqrt{1+\text{cosec}x}dx=f\left(x\right)+C$, where $C$ is a constant of integration, then $f\left(x\right)$ is equal to

1. ${\cos }^{-1}(1-2\sin x)$
2. ${\sin }^{-1}(1-2\sin x)$
3. $-2{\sin }^{-1}(1-2\sin x)$
4. ${\cos }^{-1}(1+2\sin x)$