Question

Let $x=f^{\prime \prime }\left(t\right)cost+f^{\prime }\left(t\right)sint$ and $y={-f}^{\prime \prime }\left(t\right)sint+f^{\prime }\left(t\right)cost.$ Then $\int {[{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2]}^{\frac{1}{2}}dt$ equals
(Note : $f\left(x\right),f^{\prime }\left(x\right),f^{\prime \prime }\left(x\right),f^{\prime \prime \prime }\left(x\right)>0$ )

• A. $f^{\prime }\left(t\right)+f^{\prime \prime }\left(t\right)+c$
• B. $f^{\prime \prime }\left(t\right)+f^{\prime \prime \prime }\left(t\right)+c$
• C. $f\left(t\right)+f^{\prime \prime }\left(t\right)+c$
• D. $f^{\prime }\left(t\right)-f^{\prime \prime }\left(t\right)+c$

Answer 1

The correct option is C $f\left(t\right)+f^{\prime \prime }\left(t\right)+c$

$x=f^{\prime \prime }\left(t\right)\cos t+f^{\prime }\left(t\right)\sin t\frac{dx}{dt}=f^{\prime \prime \prime }\left(t\right)\cos t-f^{\prime \prime }\left(t\right)\sin t+f^{\prime \prime }\left(t\right)\sin t+f^{\prime }\left(t\right)\cos t\frac{dx}{dt}=f^{\prime \prime \prime }\left(t\right)\cos t+f^{\prime }\left(t\right)\cos t\frac{dx}{dt}=\left(f^{\prime \prime \prime }\right(t)+f^{\prime }(t\left)\right)\cos t..........\left(1\right)y=-f^{\prime \prime }\left(t\right)\sin t+f^{\prime }\left(t\right)\cos t\frac{dy}{dt}=-f^{\prime \prime \prime }\left(t\right)\sin t-f^{\prime \prime }\left(t\right)\cos t+f^{\prime \prime }\left(t\right)\cos t-f^{\prime }\left(t\right)\sin t\frac{dy}{dt}=-f^{\prime \prime \prime }\left(t\right)\sin t-f^{\prime }\left(t\right)\sin t\frac{dy}{dt}=-\sin t\left(f^{\prime \prime \prime }\right(t)+f^{\prime }(t\left)\right)........\left(2\right)addsquaresofeq.1and2{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2={\left(f^{\prime \prime \prime }\right(t)+f^{\prime }(t\left)\right)}^2{({\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2)}^{\frac{1}{2}}=f^{\prime \prime \prime }\left(t\right)+f^{\prime }\left(t\right)integratebothsides\int {({\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2)}^{\frac{1}{2}}dt=f^{\prime \prime }\left(t\right)+f\left(t\right)+c$