Change of Variables

Q. The greatest integer less than or equal to $\underset{1}{\overset{2}{\int }}{\log }_2(x^3+1)dx+\underset{1}{\overset{{\log }_{2}9}{\int }}(2^x-1{)}^{\frac{1}{3}}dx$ is

Q. The value of the integral ${\int }_{\pi }^{0}cos(\pi -x)$ will be equal to the value of which of the following integral.

1. ${\int }_0^{\pi }cos\left(x\right)dx$
2. $-{\int }_0^{\pi }cos\left(x\right)dx$
3. ${\int }_0^{\pi }sin\left(x\right)dx$
4. $-{\int }_0^{\pi }sin\left(x\right)dx$

Q. If $I_n=\underset{0}{\overset{\pi }{\int }}\frac{1-\cos nx}{1-\cos x}dx,$ where $n$ is whole number, then

1. $I_n,I_{n+1},I_{n+2}$ are in $G.P.$
2. $I_n,I_{n+1},I_{n+2}$ are in $A.P.$
3. $I_n=n\pi$
4. $\underset{0}{\overset{\pi /2}{\int }}\frac{{\sin }^{2}n\theta }{{\sin }^2\theta }d\theta =\frac{n\pi }{2}$

Q.

If for non-zero $x$, $3f\left(x\right)+4f\left(\frac{1}{x}\right)=\frac{1}{x}-10$, then ${\int }_2^{3}f\left(x\right)dx$ is equal to

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

2. None of these
   

3. $\frac{3}{7}ln\frac{3}{2}$
   

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

Q. ${\int }_{a+c}^{b+c}f\left(x\right)\mathrm{dx}$ is equal to

1. ${\int }_a^{b}f(x-c)\mathrm{dx}$
2. ${\int }_a^{b}f\left(x\right)\mathrm{dx}$
3. ${\int }_a^{b}f(x+c)\mathrm{dx}$
4. ${\int }_{a-c}^{b-c}f\left(x\right)\mathrm{dx}$

Q. If f is a continuous function in the interval [a, b], then the value of ${\int }_0^{1}f\left(\left(b-a\right)x+a\right)\mathrm{dx}$ is equal to

1. $\left(b-a\right){\int }_a^{b}f\left(x\right)\mathrm{dx}$
2. $\left(a-b\right){\int }_a^{b}f\left(x\right)\mathrm{dx}$
3. $\frac{1}{\left(}{\int }_a^{b}f\left(x\right)\mathrm{dx}$
4. $\frac{1}{\left(}{\int }_b^{a}f\left(x\right)\mathrm{dx}$

Q. $f\left(x\right)=\left|\begin{array}{ccc}\sec x& \cos x& {\sec }^{2}x+\cot x\text{cosec}x\\ {\cos }^{2}x& {\cos }^{2}x& {\text{cosec}}^{2}x\\ 1& {\cos }^{2}x& {\cos }^{2}x\end{array}\right|$. If $\underset{0}{\overset{\pi /2}{\int }}f\left(x\right)dx=-\left(\frac{k\pi +m}{60}\right)$, then $k+m$ is equal to

Q. Let $f:[0,2]\to R$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f\left(0\right)=1.$ Let $F\left(x\right)=\underset{0}{\overset{x^2}{\int }}f\left(\sqrt{t}\right)dt,$ for $x\in [0,2]$. If $F^{\prime }\left(x\right)=f^{\prime }\left(x\right),\forall x\in (0,2),$ then $F\left(2\right)$ equals

1. $e^2-1$
2. $e^4-1$
3. $e-1$
4. $e^4$

Q. $\underset{0}{\overset{a}{\int }}\frac{dx}{x+\sqrt{a^2-x^2}}$ is

1. $\frac{a^2}{4}$
2. $\frac{\pi }{2}$
3. $\frac{\pi }{4}$
4. $\pi$

Q. If $f\left(x\right)=\underset{1}{\overset{x}{\int }}\frac{{\tan }^{-1}\left(t\right)}{t}dt\forall x\in R^{+}$, and the value of $f\left(e^2\right)-f\left(\frac{1}{e^2}\right)=\frac{k\pi }{2}$, then $k=$