Integration as Antiderivative

Q.

What does integration mean?

Q. $\int (sin2x-cos2x)dx=\frac{1}{\sqrt{2}}sin(2x-a)+c$ then a =

1. $\frac{5\pi }{4}$
2. $-\frac{5\pi }{4}$
3. $\frac{\pi }{4}$
4. $\frac{3\pi }{4}$

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

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

Q. Let $x=f^{\prime \prime }\left(t\right)\cos t+f^{\prime }\left(t\right)\sin t$ and $y=-f^{\prime \prime }\left(t\right)\sin t+f^{\prime }\left(t\right)\cos t$, then $\int {[{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2]}^{1/2}dt$ equals

1. $f^{\prime }\left(t\right)+f^{\prime \prime }\left(t\right)+c$
2. $f^{\prime \prime }\left(t\right)+f^{\prime \prime \prime }\left(t\right)+c$
3. $f\left(t\right)+f^{\prime \prime }\left(t\right)+c$
4. $f^{\prime }\left(t\right)-f^{\prime \prime }\left(t\right)+c$

Q. If $f\left(x\right)$ is invertible and twice differentiable function satisfying $f^{\prime }\left(x\right)=\stackrel{f\left(x\right)}{\underset{0}{\int }}{f}^{-1}\left(t\right)dt,\forall x\in R$ and $f^{\prime }\left(0\right)=1$, then $f^{\prime }\left(1\right)$ can be

1. $e$
2. $e^2$
3. $\frac{1}{e}$
4. $\sqrt{e}$

Q. Which of the following integral represents the summation ${\sum }^{}\left(4x^3\right)\delta x$ for the limit $\delta x\to 0$?

1. 
   $\int \left(4x\right)dx$
2. 
   $\int \left(4x^2\right)$
3. 
   $\int \left(4x^3\right)dx$
4. 
   $\int \left(4x^4\right)dx$

Q. For the differential equation $ydx+y^{2}dy=xdy,x\in R,y\left(1\right)=1,$ then the value of $y(-3)$ is:

1. $3$
2. $4$
3. $-1$
4. $-2$

Q. If $f\left(x\right)=\sin x+{\int }_0^x{f}^{\prime }\left(t\right)(2\sin t-{\sin }^{2}t)dt$ then $f\left(x\right)$ is

1. $\frac{x}{1-\sin x}$
2. $\frac{\sin x}{1-\sin x}$
3. $\frac{1-\cos x}{\cos x}$
4. $\sin x$