Change of Variables

Q. Let $f$ be a non-negative continuous function defined on $R$ such that $f\left(x\right)+f(x+\frac{1}{2})=6$ and $N=\underset{0}{\overset{6}{\int }}f\left(x\right)dx.$ Then

1. number of divisors of $N$ is $6.$
2. number of divisors of $N$ is $9.$
3. sum of the digits in $N$ is $9.$
4. sum of the digits in $N$ is $6.$

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

Q.

Find the domain of $f\left(x\right)=cotx+co{t}^{-1}x$

Q.

Which transformation causes the described change in the graph of function $y=\cos \left(x\right)$?

The transformation results in horizontal stretch.

1. $\frac{2}{7}\cos \left(x\right)$

2. $-3\cos \left(x\right)$

3. $\cos \left(7x\right)$

4. $\cos \left(\frac{x}{3}\right)$

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

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

4. None of these
   

Q. Let $f\left(x\right)=\underset{0}{\overset{x}{\int }}g\left(t\right)dt$, where $g$ is a non-zero even function. If $f(x+5)=g\left(x\right)$, then $\underset{0}{\overset{x}{\int }}f\left(t\right)dt$ equals :

1. $\underset{x+5}{\overset{5}{\int }}g\left(t\right)dt$
2. $\underset{5}{\overset{x+5}{\int }}g\left(t\right)dt$
3. $2\underset{5}{\overset{x+5}{\int }}g\left(t\right)dt$
4. $5\underset{x+5}{\overset{5}{\int }}g\left(t\right)dt$

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. 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 $k=$

Q. Which of the following is the total general solution of the equation $2^{\frac{1}{1-|\cos x|}}=2^2$ ?

1. $x=\{n\pi \pm \frac{\pi }{6},n\in Z\}$
2. $x=\{n\pi \pm \frac{\pi }{3},n\in Z\}$
3. $x=\{n\pi \pm \frac{\pi }{4},n\in Z\}$
4. All of the above