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A particle moves along the $x$-axis from $x=0$ to $x=1$ under the influence of the force $F=(1+x^2)^{-1}$.Find the work done by this force.

$\begin{array}{1 1}\large\frac{\pi}{4}\\\large\frac{\pi}{2}\\\large\frac{\pi}{2}\\\pi\end{array}$

Answer : $\large\frac{\pi}{4}$
The displacement variable is $x$ (not s) we can use the formula $dW = Fdx=\large\frac{dx}{1+x^2}$.Then
$W=\int dW$
$\;\;\;\;=\int\limits_0^{1} \large\frac{dx}{1+x^2}$
$\;\;\;\;=\int\limits_0^{\Large\frac{\pi}{4}} \large\frac{\sec^2\theta}{1+\tan^2\theta}$$d\theta$
$\;\;\;\;=\int\limits_0^{\Large\frac{\pi}{4}} d\theta$
$\;\;\;\;=\large\frac{\pi}{4}$

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