Integration

Q. If $\int \frac{x^2+4}{x^4+16}dx=\frac{1}{k}{\tan }^{-1}\left(\frac{x^2-4}{kx}\right)+C$, then $k^2=$(where $k$ is fixed constant and $C$ is integration constant)

Q. If $(2+\sqrt{3}{)}^n=I+f,n\in N$, where $I$ is integral part and $f$ is fractional part, then $(I+f)(1-f)$ is

1. $0$
2. $1$
3. $n$
4. $-1$

Q. $\int \sqrt{1+3x-x^2}dx$ is equal to
(where $C$ is integration constant)

1. $\frac{2x-3}{4}\sqrt{1+3x-x^2}+\frac{13}{8}\ln |2x-3+\sqrt{1+3x-x^2}|+C$
2. $\frac{2x-3}{4}\sqrt{1+3x-x^2}+\frac{13}{8}{\sin }^{-1}\left[\frac{2x-3}{\sqrt{13}}\right]+C$
3. $\frac{x}{2}\sqrt{1+3x-x^2}+\frac{13}{8}{\sin }^{-1}\left[\frac{2x}{\sqrt{13}}\right]+C$
4. $\frac{x}{2}\sqrt{1+3x-x^2}-\frac{13}{8}\ln |2x-3+\sqrt{1+3x-x^2}|+C$

Q. If $(2\le r\le n),$ then ${}^n{C}_r+2{\cdot }^n{C}_{r+1}{+}^n{C}_{r+2}$ is equal to

1. $2{\cdot }^n{C}_{r+2}$
2. ${}^{n+1}{C}_{r+1}$
3. ${}^{n+2}{C}_{r+2}$
4. ${}^{n+1}{C}_r$

Q.

If n is even and the value of ${}^n{C}_r$ is maximum, then r =

1. 

2. 

3. 

4. n/4

Q. Let $f:[0,1]\to [0,1]$ be a continuous function. Then

1. $f\left(x\right)+x=0$ for at least one $x$ such that $0\le x\le 1$
2. $f\left(x\right)=x$ for at least one $0\le x\le 1$
3. none of these
4. $f\left(x\right)$ will be differentiable in $[0,1]$

Q. $\int (2x-3\cos x+e^x)dx$ is equal to (where $C$ is the constant of integration)

1. $x^2-3\sin x+e^x+C$
2. $x^2+3\sin x+e^x+C$
3. $x^2+3\sin x+e^{-x}+C$
4. $x^3+3\sin x+e^x+C$

Q.

${}^{n+1}{C}_{n+1}=?n\in N$

1. 

2. 

3. 

4. 

Q. $\int \sqrt{\frac{1+x^2}{x^2-x^4}}dx$ equals to
(where $C$ is constant of integration)

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

Q. If $I_n={\int }_0^{\pi /4}ta{n}^{n}xdx$, then $I_2+I_4,I_3+I_5,I_4+I_6,I_5+I_7$, ....... are in

1. A.P
2. G.P
3. none of these
4. H.P

Q. $\int \frac{2x+5}{\sqrt{7-6x-x^2}}dx=A\sqrt{7-6x-x^2}+B{\sin }^{-1}\left(\frac{x+3}{4}\right)+C$ (where C is a constant of integration), then the value of $B-A$ is equal to

Q. The value of integral $\int \frac{dx}{x(\ln x{)}^3}$ is
(where $C$ is constant of integration)

1. $\frac{1}{(\ln x{)}^3}+C$
2. $-\frac{1}{2(\ln x{)}^2}+C$
3. $-\frac{1}{3(\ln x{)}^3}+C$
4. $-\ln \left(x\right)+C$

Q. The value of $\frac{{}^n{C}_0}{1\cdot 2}-\frac{{}^n{C}_1}{2\cdot 3}+\frac{{}^n{C}_2}{3\cdot 4}+\cdots +(-1{)}^n\frac{{}^n{C}_n}{(n+1)\cdot (n+2)}$ is

1. $\frac{1}{n+2}$
2. $\frac{1}{n+1}$
3. $\frac{1}{n+3}$
4. $\frac{1}{(n+2)(n+1)}$

Q. Sum of series $S=\sum _{r=0}^n\frac{3^{r+4}\cdot {}^n{C}_r}{{}^{r+4}{C}_4}+\sum _{r=0}^3\frac{3^r\cdot {}^{n+4}{C}_r}{{}^{n+4}{C}_4}$

1. $\frac{4^{n+4}}{{}^{n+3}{C}_n+{}^{n+3}{C}_{n-1}}$
2. $4^{n+4}\left({}^{2n}{C}_4\right)$
3. $\frac{4^{n+4}}{{}^{n+4}{C}_4}$
4. $\frac{3^{n+4}+2^{n+4}}{{}^{n+4}{C}_4}$

Q. $\int \frac{e^x}{\sqrt{4-e^{2x}}}dx$ equals
(where $C$ is constant of integration)

1. ${\sin }^{-1}\left(\frac{e^x}{2}\right)+C$
2. $2{\sin }^{-1}\left(e^x\right)+C$
3. $\frac{1}{2}\ln \left|\frac{2+e^x}{2-e^x}\right|+C$
4. $\frac{1}{2}{\sin }^{-1}\left(e^x\right)+C$

Q. $\int \frac{1}{4+9x^2}dx$ is equal to
(where $C$ is constant of integration)

1. $\frac{1}{6}{\tan }^{-1}\frac{3x}{2}+C$
2. $\frac{1}{3}{\tan }^{-1}\frac{6x}{2}+C$
3. $\frac{1}{2}{\tan }^{-1}\frac{5x}{2}+C$
4. $\frac{1}{3}{\tan }^{-1}\frac{7x}{2}+C$

Q. $\int \frac{1}{\sqrt{1-(2-\sqrt{3}x{)}^2}}dx$ is equal to
(where $C$ is constant of integration)

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

Q. $\int e^x\sqrt{e^{2x}+1}dx$ is equal to
(where $C$ is integration constant)

1. $\frac{e^x}{2}\sqrt{e^{2x}+1}+\frac{1}{2}\ln |e^x+\sqrt{e^{2x}+1}|+C$
2. $\frac{1}{2}\ln |e^x+\sqrt{e^{2x}+1}|+C$
3. $\frac{e^x}{2}\sqrt{e^x+1}+\frac{1}{2}\ln |e^x+\sqrt{e^{2x}+1}|+C$
4. $\frac{1}{4}\ln \left|\frac{e^x+\sqrt{e^x+1}}{e^x-\sqrt{e^x+1}}\right|+C$

Q. If $\int \frac{x^4+1}{x^6+1}dx={\tan }^{-1}f\left(x\right)-\frac{2}{3}{\tan }^{-1}g\left(x\right)+C$ for constant of integration $C,$ then $f\left(x\right)-g\left(x\right)$ is

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

Q. The value of integral $\int \frac{dx}{(x+2)\sqrt{x+1}}$ is (where $C$ is integration constant)

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

Q. Evaluate $\underset{x\to 0}{lim}\frac{x-ta{n}^{-1}x}{x^3}$ using series expansion

1. $\frac{1}{3}$
2. $\frac{4}{7}$
3. $\frac{3}{2}$
4. $\frac{2}{3}$

Q. Evaluate the definite integral ${\int }_0^{\frac{\pi }{4}}(2{\sec }^{2}x+x^3+2)dx$

Q. $\int \frac{x^2+1}{x^4+x^2+1}dx$ is equal to (where $C$ is integration constant)

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

Q. The value of $\frac{{}^n{C}_0}{1\cdot 2}-\frac{{}^n{C}_1}{2\cdot 3}+\frac{{}^n{C}_2}{3\cdot 4}+\cdots +(-1{)}^n\frac{{}^n{C}_n}{(n+1)\cdot (n+2)}$ is

1. $\frac{1}{n+2}$
2. $\frac{1}{n+1}$
3. $\frac{1}{n+3}$
4. $\frac{1}{(n+2)(n+1)}$

Q. $\int \frac{1}{4+9x^2}dx$ is equal to
(where $C$ is constant of integration)

1. $\frac{1}{6}{\tan }^{-1}\frac{3x}{2}+C$
2. $\frac{1}{3}{\tan }^{-1}\frac{6x}{2}+C$
3. $\frac{1}{2}{\tan }^{-1}\frac{5x}{2}+C$
4. $\frac{1}{3}{\tan }^{-1}\frac{7x}{2}+C$

Q. If $(2+\sqrt{3}{)}^n=I+f,n\in N$, where $I$ is integral part and $f$ is fractional part, then $(I+f)(1-f)$ is

1. $1$
2. $0$
3. $n$
4. $-1$